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

<span class="kw">set </span><font color="Maroon" title="c3">XS</font> =  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c4">YS</font> =  <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c5">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/>
<a NAME="E1:1_1_1"/><span class="lab"><font color="Green" title="E2">A2</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="newton.html#K15" title="NEWTON:func.15">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 3
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <span class="kw">means </span>( ( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies $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> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</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 $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> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</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"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies ( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; $3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) ) &amp; ( (  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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Olive" title="b1">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) 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="E2:1_1_1"/><span class="lab"><font color="Green" title="E3">A3</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T49" title="JORDAN1H:th.49">JORDAN1H:49</a></span>;<br/>
<a NAME="E3:1_1_1"/><span class="lab"><font color="Green" title="E4">A4</font></span>: 
 <a href="pre_topc.html#G1" title="PRE_TOPC:aggr.1">TopStruct</a>(#  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>, the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pcomps_1.html#K3" title="PCOMPS_1:func.3">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="euclid.html#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="euclid.html#D8" title="EUCLID:def.8">EUCLID:def 8</a></span>;<br/>
<a NAME="E4:1_1_1"/><span class="lab"><font color="Green" title="E5">A5</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="ref" href="jordan11.html#T8" title="JORDAN11:th.8">JORDAN11:8</a></span>;<br/>
<a NAME="E5:1_1_1"/><span class="lab"><font color="Green" title="E6">A6</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></span>;<br/>
<a NAME="E6:1_1_1"/><span class="lab"><font color="Green" title="E7">A7</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <br/>  for <font color="Olive" title="b2">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b3">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Olive" title="b2">x</font>,<font color="Olive" title="b3">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   ex <font color="Olive" title="b2">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c7">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_1_1"/>
( <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or ( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c7">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="c6">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c7">x</font> is  not   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T25" title="NAT_1:th.25">NAT_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E8">A8</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span>
<span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">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> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/><span class="kw">thus </span><a NAME="E2:1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">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> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_1"><span class="lab"><font color="Green" title="E8">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E8">A9</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span>
<span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</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"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">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> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</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"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>]</span><br/><span class="kw">thus </span><a NAME="E2:1_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">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> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</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"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_2"><span class="lab"><font color="Green" title="E8">A9</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E8">A10</font></span>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E9">A11</font></span>: 
<font color="Maroon" title="c7">x</font> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">x</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E9">A11</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E4:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_1_1_3_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E10">A12</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1"/>
( ( <font color="Maroon" title="c8">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  not <font color="Maroon" title="c8">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E11">A13</font></span>: 
( <font color="Maroon" title="c8">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E12">A14</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A15</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E14">A16</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E15">A17</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">i1</font>, <font color="Maroon" title="c10">j1</font>, <font color="Maroon" title="c11">i2</font>, <font color="Maroon" title="c12">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E16">A18</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E17">A19</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font>)
 <span class="kw">and </span><br/><a NAME="E9:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E18">A20</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E19">A21</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">and </span><br/><a NAME="E11:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E20">A22</font></span>: 
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/><a NAME="E12:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E21">A23</font></span>: 
<font color="Maroon" title="c9">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E13:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E22">A24</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E14:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E23">A25</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E15:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E24">A26</font></span>: 
<font color="Maroon" title="c10">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E16:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E25">A27</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E17:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E26">A28</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E18:1_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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T35" title="NAT_D:th.35">NAT_D:35</a></span>;<br/><a NAME="E19:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A29</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E20:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E28">A30</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E21:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A31</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/><a NAME="E22:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E30">A32</font></span>: 
<font color="Maroon" title="c11">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E23:1_1_1_1_1_3_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A33</font></span>: 
<font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/><span class="kw">thus </span><a NAME="E24:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1"/>
( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E32">A34</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E20">A22</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E33">A35</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c13">f1</font> = <font color="Maroon" title="c8">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="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Olive" title="b1">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c12">j2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E34">A36</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c10">j1</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E33">A35</font></span></a>, <a class="ref" href="gobrd13.html#T21" title="GOBRD13:th.21">GOBRD13:21</a></span>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c10">j1</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T232" title="XREAL_1:th.232">XREAL_1:232</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E15:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E24">A26</font></span></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) implies ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E35">A37</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A38</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E37">A39</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A40</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )</span><br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A41</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A39</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A42</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A40</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A43</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A38</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A38</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A42</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E17:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E20:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E23:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A15</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E33">A35</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E35">A37</font></span></a>, <a class="txt" href="#E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A41</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A43</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A36</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E32">A34</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E33">A44</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c13">f1</font> = <font color="Maroon" title="c8">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="c11">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c11">i2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Olive" title="b1">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E34">A45</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A46</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E20:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E35">A46</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c9">i1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E33">A44</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="ref" href="jordan8.html#T15" title="JORDAN8:th.15">JORDAN8:15</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) implies ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E35">A47</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E36">A48</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E37">A49</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E38">A50</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )</span><br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E39">A51</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E37">A49</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><span class="lab"><font color="Green" title="E40">A52</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A50</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A53</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E36">A48</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E36">A48</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E40">A52</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/>
( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E14:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="txt" href="#E22:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="txt" href="#E13:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A15</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E33">A44</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E35">A47</font></span></a>, <a class="txt" href="#E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E39">A51</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E41">A53</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><span class="lab"><font color="Green" title="E34">A45</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E32">A34</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E33">A54</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c13">f1</font> = <font color="Maroon" title="c8">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="c11">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c11">i2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Olive" title="b1">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E34">A55</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/>
<font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E33">A54</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c11">i2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T232" title="XREAL_1:th.232">XREAL_1:232</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="ref" href="jordan8.html#T17" title="JORDAN8:th.17">JORDAN8:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) implies ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E35">A56</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E36">A57</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E37">A58</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E38">A59</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )</span><br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E39">A60</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E37">A58</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><span class="lab"><font color="Green" title="E40">A61</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E38">A59</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A62</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E36">A57</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E36">A57</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E40">A61</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/>
( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E14:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="txt" href="#E22:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="txt" href="#E21:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E29">A31</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A15</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E33">A54</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E35">A56</font></span></a>, <a class="txt" href="#E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E39">A60</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E41">A62</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><span class="lab"><font color="Green" title="E34">A55</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1"/>
<font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1"/>
<font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E32">A34</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E33">A63</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c13">f1</font> = <font color="Maroon" title="c8">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="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Olive" title="b1">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c12">j2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E34">A64</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A65</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"><span class="lab"><font color="Green" title="E35">A65</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c12">j2</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E33">A63</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E20:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="ref" href="jordan8.html#T16" title="JORDAN8:th.16">JORDAN8:16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) implies ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><span class="lab"><font color="Green" title="E35">A66</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><span class="lab"><font color="Green" title="E36">A67</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><span class="lab"><font color="Green" title="E37">A68</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><span class="lab"><font color="Green" title="E38">A69</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )</span><br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><span class="lab"><font color="Green" title="E39">A70</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E37">A68</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><span class="lab"><font color="Green" title="E40">A71</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E38">A69</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A72</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E36">A67</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E36">A67</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E40">A71</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/>
( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E17:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E20:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E16:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A15</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E33">A63</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E35">A66</font></span></a>, <a class="txt" href="#E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E39">A70</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E41">A72</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><span class="lab"><font color="Green" title="E34">A64</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1"/>
<font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1"/>
<font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E32">A34</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E32">A73</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E20">A22</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E33">A74</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c13">f1</font> = <font color="Maroon" title="c8">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="c11">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c11">i2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Olive" title="b1">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E34">A75</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A76</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E20:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E35">A76</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</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>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A73</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E33">A74</font></span></a>, <a class="ref" href="gobrd13.html#T34" title="GOBRD13:th.34">GOBRD13:34</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="ref" href="jordan8.html#T15" title="JORDAN8:th.15">JORDAN8:15</a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) implies ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E35">A77</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A78</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E37">A79</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A80</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) )</span><br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A81</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A79</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A82</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A80</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A83</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A78</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A78</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A82</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/>
( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E14:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="txt" href="#E22:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="txt" href="#E13:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A15</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E33">A74</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E35">A77</font></span></a>, <a class="txt" href="#E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A81</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A83</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A75</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A73</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E33">A84</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c13">f1</font> = <font color="Maroon" title="c8">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="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Olive" title="b1">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c12">j2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E34">A85</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A86</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E35">A86</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c10">j1</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A73</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E33">A84</font></span></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E15:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E24">A26</font></span></a>, <a class="ref" href="jordan8.html#T16" title="JORDAN8:th.16">JORDAN8:16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) implies ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E35">A87</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E36">A88</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E37">A89</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E38">A90</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) )</span><br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E39">A91</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E37">A89</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><span class="lab"><font color="Green" title="E40">A92</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A90</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A93</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E36">A88</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E36">A88</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E40">A92</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/>
( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E17:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E20:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E16:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A15</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E33">A84</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E35">A87</font></span></a>, <a class="txt" href="#E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E39">A91</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E41">A93</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><span class="lab"><font color="Green" title="E34">A85</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A73</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E33">A94</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c13">f1</font> = <font color="Maroon" title="c8">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="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Olive" title="b1">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c12">j2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E34">A95</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A73</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E33">A94</font></span></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></span>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T232" title="XREAL_1:th.232">XREAL_1:232</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E15:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E24">A26</font></span></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) implies ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E35">A96</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E36">A97</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E37">A98</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E38">A99</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) )</span><br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E39">A100</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E37">A98</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E40">A101</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E38">A99</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A102</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E36">A97</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E36">A97</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E40">A101</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/>
( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E17:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E20:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E23:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A15</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E33">A94</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E35">A96</font></span></a>, <a class="txt" href="#E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E39">A100</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E41">A102</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E34">A95</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1"/>
<font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1"/>
<font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A73</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E33">A103</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c13">f1</font> = <font color="Maroon" title="c8">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="c11">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c11">i2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Olive" title="b1">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E34">A104</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/>
<font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c9">i1</font>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A73</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E33">A103</font></span></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></span>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c9">i1</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T232" title="XREAL_1:th.232">XREAL_1:232</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E21">A23</font></span></a>, <a class="ref" href="jordan8.html#T17" title="JORDAN8:th.17">JORDAN8:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) implies ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E35">A105</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E36">A106</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E37">A107</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E38">A108</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) )</span><br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E39">A109</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E37">A107</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E40">A110</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E38">A108</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A111</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E36">A106</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E36">A106</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E40">A110</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/>
( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E14:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="txt" href="#E22:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="txt" href="#E21:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E29">A31</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A15</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E33">A103</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E35">A105</font></span></a>, <a class="txt" href="#E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E39">A109</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E41">A111</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E34">A104</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1"/>
<font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1"/>
<font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E32">A73</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E32">A112</font></span>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E20">A22</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E33">A113</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c13">f1</font> = <font color="Maroon" title="c8">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="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Olive" title="b1">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c12">j2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E34">A114</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A115</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E35">A115</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>,<font color="Maroon" title="c12">j2</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E32">A112</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E33">A113</font></span></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E20:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="ref" href="jordan8.html#T16" title="JORDAN8:th.16">JORDAN8:16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) implies ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E35">A116</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E36">A117</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E37">A118</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E38">A119</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )</span><br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E39">A120</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E37">A118</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A121</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E38">A119</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A122</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A117</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E36">A117</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A121</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/>
( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E17:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E20:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E16:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E25">A27</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A15</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E33">A113</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E35">A116</font></span></a>, <a class="txt" href="#E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E39">A120</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A122</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><span class="lab"><font color="Green" title="E34">A114</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E32">A112</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E33">A123</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c13">f1</font> = <font color="Maroon" title="c8">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="c11">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c11">i2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Olive" title="b1">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E34">A124</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E32">A112</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E33">A123</font></span></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></span>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c11">i2</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T232" title="XREAL_1:th.232">XREAL_1:232</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="ref" href="jordan8.html#T17" title="JORDAN8:th.17">JORDAN8:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) implies ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E35">A125</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E36">A126</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E37">A127</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E38">A128</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )</span><br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E39">A129</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E37">A127</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><span class="lab"><font color="Green" title="E40">A130</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E38">A128</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A131</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E36">A126</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E36">A126</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E40">A130</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/>
( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E14:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="txt" href="#E22:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="txt" href="#E21:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E29">A31</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A15</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E33">A123</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E35">A125</font></span></a>, <a class="txt" href="#E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E39">A129</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E41">A131</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><span class="lab"><font color="Green" title="E34">A124</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1"/>
<font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1"/>
<font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E32">A112</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E33">A132</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c13">f1</font> = <font color="Maroon" title="c8">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="c11">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c11">i2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Olive" title="b1">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E34">A133</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/>
<font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A134</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/>
<font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E20:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E35">A134</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</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>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E32">A112</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E33">A132</font></span></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="ref" href="jordan8.html#T15" title="JORDAN8:th.15">JORDAN8:15</a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) implies ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><span class="lab"><font color="Green" title="E35">A135</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><span class="lab"><font color="Green" title="E36">A136</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><span class="lab"><font color="Green" title="E37">A137</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><span class="lab"><font color="Green" title="E38">A138</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )</span><br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><span class="lab"><font color="Green" title="E39">A139</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E37">A137</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><span class="lab"><font color="Green" title="E40">A140</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E38">A138</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A141</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E36">A136</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E36">A136</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E40">A140</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/>
( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E14:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="txt" href="#E22:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E30">A32</font></span></a>, <a class="txt" href="#E13:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E22">A24</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A15</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E33">A132</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E35">A135</font></span></a>, <a class="txt" href="#E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E39">A139</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E41">A141</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><span class="lab"><font color="Green" title="E34">A133</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1"/>
<font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1"/>
<font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E32">A112</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E33">A142</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c13">f1</font> = <font color="Maroon" title="c8">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="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c14">i</font> = <font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Olive" title="b1">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span><font color="Maroon" title="c15">j</font> = <font color="Maroon" title="c12">j2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="kw">now </span></a><div class="add"><div><span class="lab"><font color="Green" title="E34">A143</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a>, <a class="ref" href="nat_d.html#T36" title="NAT_D:th.36">NAT_D:36</a></span>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E23">A25</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default">1 <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E15">A17</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E12">A14</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E32">A112</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E33">A142</font></span></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></span>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/><span class="kw">then </span>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T232" title="XREAL_1:th.232">XREAL_1:232</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E20:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="ref" href="jordan8.html#T18" title="JORDAN8:th.18">JORDAN8:18</a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) &amp;  not <!--HUMANRECFAILED-->( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) implies ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><span class="lab"><font color="Green" title="E35">A144</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><span class="lab"><font color="Green" title="E36">A145</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><span class="lab"><font color="Green" title="E37">A146</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><span class="lab"><font color="Green" title="E38">A147</font></span>: 
<font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )</span><br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><span class="lab"><font color="Green" title="E39">A148</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E27">A29</font></span></a>, <a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E37">A146</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><span class="lab"><font color="Green" title="E40">A149</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E14">A16</font></span></a>, <a class="txt" href="#E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E38">A147</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A150</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E36">A145</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A20</font></span></a>, <a class="txt" href="#E10:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E19">A21</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E36">A145</font></span></a>, <a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E40">A149</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/>
( ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c16">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font> &amp; <font color="Maroon" title="c17">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c13">f1</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c19">j29</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A18</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A19</font></span></a>, <a class="txt" href="#E17:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E26">A28</font></span></a>, <a class="txt" href="#E20:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E28">A30</font></span></a>, <a class="txt" href="#E23:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E31">A33</font></span></a>, <a class="txt" href="#E3:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E13">A15</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E33">A142</font></span></a>, <a class="txt" href="#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E35">A144</font></span></a>, <a class="txt" href="#E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E39">A148</font></span></a>, <a class="txt" href="#E8:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E41">A150</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><span class="lab"><font color="Green" title="E34">A143</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a>, <a class="ref" href="matrix_1.html#T36" title="MATRIX_1:th.36">MATRIX_1:36</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1"/>
<font color="Maroon" title="c13">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1"/>
<font color="Maroon" title="c13">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">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">i</font>,<font color="Maroon" title="c15">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_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="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c13">f1</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E11">A13</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E32">A112</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E11">A151</font></span>: 
(  not <font color="Maroon" title="c8">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span>
<font color="Maroon" title="c8">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]</span><br/><span class="kw">thus </span><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E10">A12</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E11">A151</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E10">A152</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]</span><br/><span class="kw">thus </span><a NAME="E2:1_1_1_1_1_3_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_3"><span class="lab"><font color="Green" title="E8">A10</font></span></a>, <a class="txt" href="#E1:1_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E10">A152</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E8">A153</font></span>: 
( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <font color="Maroon" title="c7">x</font> is  not   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">y</font>]</span><br/><div class="add"><span class="kw">take </span>
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]</span><br/><span class="kw">thus </span><a NAME="E2:1_1_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font>,<font color="Maroon" title="c7">x</font>, <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_1_1_4"><span class="lab"><font color="Green" title="E8">A153</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c6">F</font> being   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a><span class="kw"> such that </span><br/><a NAME="E8:1_1_1"/><span class="lab"><font color="Green" title="E8">A154</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">and </span><br/><a NAME="E9:1_1_1"/><span class="lab"><font color="Green" title="E9">A155</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">and </span><br/><a NAME="E10:1_1_1"/><span class="lab"><font color="Green" title="E10">A156</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c6">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="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="recdef_1.html#S1" title="RECDEF_1:sch.1">RECDEF_1:sch 1</a>(<a class="txt" href="#E6:1_1_1"><span class="lab"><font color="Green" title="E7">A7</font></span></a>);<br/></span>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span><font color="Maroon" title="c6">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="E11:1_1_1"/><span class="lab"><font color="Green" title="E11">A157</font></span>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K6" title="FINSEQ_1:func.6">&lt;*&gt;</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
;<br/>
<a NAME="E12:1_1_1"/><span class="lab"><font color="Green" title="E12">A158</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_1_2"/><span class="lab"><font color="Green" title="E13">A159</font></span>: 
<font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_2_1"/>
( <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T25" title="NAT_1:th.25">NAT_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_2_1_1"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_2_1_2"/>
<font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_2_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_2_1_3"/><span class="lab"><font color="Green" title="E14">A160</font></span>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:1_1_1_2_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_1_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">k</font> as   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_2"><span class="lab"><font color="Green" title="E13">A159</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_2_1_3_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font> or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_2_1_3_1_1_1"/><span class="lab"><font color="Green" title="E15">A161</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:1_1_1_2_1_3_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_1_3_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_3_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_2_1_3_1_1_1_1_1"/>
( ( <font color="Maroon" title="c8">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  not <font color="Maroon" title="c8">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_3_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_2_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E16">A162</font></span>: 
( <font color="Maroon" title="c8">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_2_1_3_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A163</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_2_1_3"><span class="lab"><font color="Green" title="E14">A160</font></span></a>, <a class="txt" href="#E1:1_1_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E15">A161</font></span></a></span>;<br/><a NAME="E3:1_1_1_2_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E18">A164</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_2_1_3"><span class="lab"><font color="Green" title="E14">A160</font></span></a>, <a class="txt" href="#E1:1_1_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E15">A161</font></span></a>, <a class="txt" href="#E1:1_1_1_2_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A162</font></span></a></span>;<br/><a NAME="E4:1_1_1_2_1_3_1_1_1_1_1_1"/>
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c6">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_2_1_3"><span class="lab"><font color="Green" title="E14">A160</font></span></a>, <a class="txt" href="#E1:1_1_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E15">A161</font></span></a>, <a class="txt" href="#E1:1_1_1_2_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E16">A162</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:1_1_1_2_1_3_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_2_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E18">A164</font></span></a>, <a class="txt" href="#E2:1_1_1_2_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E17">A163</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_3_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_2_1_3_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E16">A165</font></span>: 
(  not <font color="Maroon" title="c8">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c8">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="c8">f</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_2_1_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c8">f</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 ;<br/><span class="kw">hence </span><a NAME="E3:1_1_1_2_1_3_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_2_1_3"><span class="lab"><font color="Green" title="E14">A160</font></span></a>, <a class="txt" href="#E1:1_1_1_2_1_3_1_1_1"><span class="lab"><font color="Green" title="E15">A161</font></span></a>, <a class="txt" href="#E1:1_1_1_2_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E16">A165</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_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="c8">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c7">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_2_1_3_1_1_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c7">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E11:1_1_1"><span class="lab"><font color="Green" title="E11">A157</font></span></a>, <a class="txt" href="#E1:1_1_1_2_1_3"><span class="lab"><font color="Green" title="E14">A160</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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

</div><span class="kw">end;</span></div>
<a NAME="E13:1_1_1"/><span class="lab"><font color="Green" title="E13">A166</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1"><span class="lab"><font color="Green" title="E9">A155</font></span></a>, <a class="txt" href="#E11:1_1_1"><span class="lab"><font color="Green" title="E11">A157</font></span></a></span>;<br/>
<a NAME="E14:1_1_1"/><span class="lab"><font color="Green" title="E14">A167</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E13:1_1_1"><span class="lab"><font color="Green" title="E13">A166</font></span></a>, <a class="txt" href="#E12:1_1_1"><span class="lab"><font color="Green" title="E12">A158</font></span></a>);<br/></span>
<a NAME="E15:1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">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="1_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">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="c6">F</font> or <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_1_3"/>
<font color="Maroon" title="c7">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="c6">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span><br/>

<a NAME="E2:1_1_1_3"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/>( <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">F</font> &amp; <font color="Maroon" title="c6">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="c7">y</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D3" title="FUNCT_1:def.3">FUNCT_1:def 3</a></span>;<br/>
<a NAME="E3:1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">y</font> is   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1"><span class="lab"><font color="Green" title="E8">A154</font></span></a>, <a class="txt" href="#E14:1_1_1"><span class="lab"><font color="Green" title="E14">A167</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:1_1_1_3"/>
<font color="Maroon" title="c7">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">F</font> = <font color="Maroon" title="c6">F</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>,<span class="p1">(<span class="default"> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1"><span class="lab"><font color="Green" title="E8">A154</font></span></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span> <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="E17:1_1_1"/><span class="lab"><font color="Green" title="E15">A168</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">k</font>] implies <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_1_4"/><span class="lab"><font color="Green" title="E16">A169</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:1_1_1_4"/><span class="lab"><font color="Green" title="E17">A170</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font>,<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_4_1"/>
( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T25" title="NAT_1:th.25">NAT_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_4_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_4_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_4"><span class="lab"><font color="Green" title="E17">A170</font></span></a>, <a class="ref" href="finseq_1.html#T39" title="FINSEQ_1:th.39">FINSEQ_1:39</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_4_1_2"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_4_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_4"><span class="lab"><font color="Green" title="E17">A170</font></span></a>, <a class="ref" href="finseq_1.html#T44" title="FINSEQ_1:th.44">FINSEQ_1:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_4_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_4_1_3"/><span class="lab"><font color="Green" title="E18">A171</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">thus </span><a NAME="E2:1_1_1_4_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_4_1_3_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_4_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_4_1_3_1_1"/>
( ( <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  not <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_4_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_4_1_3_1_1_1"/><span class="lab"><font color="Green" title="E19">A172</font></span>: 
( <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_4_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E20">A173</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_4"><span class="lab"><font color="Green" title="E16">A169</font></span></a>, <a class="txt" href="#E1:1_1_1_4_1_3"><span class="lab"><font color="Green" title="E18">A171</font></span></a></span>;<br/><a NAME="E3:1_1_1_4_1_3_1_1_1"/><span class="lab"><font color="Green" title="E21">A174</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_4"><span class="lab"><font color="Green" title="E16">A169</font></span></a>, <a class="txt" href="#E1:1_1_1_4_1_3"><span class="lab"><font color="Green" title="E18">A171</font></span></a>, <a class="txt" href="#E1:1_1_1_4_1_3_1_1_1"><span class="lab"><font color="Green" title="E19">A172</font></span></a></span>;<br/><a NAME="E4:1_1_1_4_1_3_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_4"><span class="lab"><font color="Green" title="E16">A169</font></span></a>, <a class="txt" href="#E1:1_1_1_4_1_3"><span class="lab"><font color="Green" title="E18">A171</font></span></a>, <a class="txt" href="#E1:1_1_1_4_1_3_1_1_1"><span class="lab"><font color="Green" title="E19">A172</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E5:1_1_1_4_1_3_1_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_4"><span class="lab"><font color="Green" title="E16">A169</font></span></a>, <a class="txt" href="#E3:1_1_1_4_1_3_1_1_1"><span class="lab"><font color="Green" title="E21">A174</font></span></a>, <a class="txt" href="#E2:1_1_1_4_1_3_1_1_1"><span class="lab"><font color="Green" title="E20">A173</font></span></a>, <a class="ref" href="finseq_2.html#T16" title="FINSEQ_2:th.16">FINSEQ_2:16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_4_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_4_1_3_1_1_2"/>
(  not <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_4_1_3_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (1,1)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_4"><span class="lab"><font color="Green" title="E16">A169</font></span></a>, <a class="txt" href="#E1:1_1_1_4_1_3"><span class="lab"><font color="Green" title="E18">A171</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:1_1_1_4_1_3_1_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_4"><span class="lab"><font color="Green" title="E16">A169</font></span></a>, <a class="ref" href="finseq_2.html#T16" title="FINSEQ_2:th.16">FINSEQ_2:16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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

</div><span class="kw">end;</span></div>
<span class="kw">defpred </span><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span>( <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1 <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; (  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1</span>)</span> holds <br/>(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="E18:1_1_1"/><span class="lab"><font color="Green" title="E16">A175</font></span>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1"><span class="lab"><font color="Green" title="E9">A155</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></span>;<br/>
<a NAME="E19:1_1_1"/><span class="lab"><font color="Green" title="E17">A176</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E18:1_1_1"><span class="lab"><font color="Green" title="E16">A175</font></span></a>, <a class="txt" href="#E17:1_1_1"><span class="lab"><font color="Green" title="E15">A168</font></span></a>);<br/></span>
<a NAME="E20:1_1_1"/><span class="lab"><font color="Green" title="E18">A177</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T49" title="JORDAN1H:th.49">JORDAN1H:49</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E21:1_1_1"/><span class="lab"><font color="Green" title="E19">A178</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c8">k</font>] implies <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_5"/><span class="lab"><font color="Green" title="E20">A179</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5"/><span class="lab"><font color="Green" title="E21">A180</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> holds <br/>(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E3:1_1_1_5"/><span class="lab"><font color="Green" title="E22">A181</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/>
<a NAME="E4:1_1_1_5"/><span class="lab"><font color="Green" title="E23">A182</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_5_1"/>
( <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T25" title="NAT_1:th.25">NAT_1:25</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_1"/><span class="lab"><font color="Green" title="E24">A183</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_5_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A184</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E26">A185</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) ) )</span><br/><span class="kw">assume </span><a NAME="E1:1_1_1_5_1_1_1"/><span class="lab"><font color="Green" title="E27">A186</font></span>: 
<font color="Maroon" title="c9">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) )</span><br/><a NAME="E2:1_1_1_5_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A187</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">l</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:1_1_1_5_1_1_1"/>
<font color="Maroon" title="c9">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_1"><span class="lab"><font color="Green" title="E24">A183</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_1_1"><span class="lab"><font color="Green" title="E27">A186</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:1_1_1_5_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_5_1_1_1"><span class="lab"><font color="Green" title="E28">A187</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E5:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_1"><span class="lab"><font color="Green" title="E25">A184</font></span></a>, <a class="ref" href="finseq_4.html#T16" title="FINSEQ_4:th.16">FINSEQ_4:16</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies  for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font>) holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">i2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_5_1_1_2"/><span class="lab"><font color="Green" title="E27">A188</font></span>: 
<font color="Maroon" title="c9">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_1_2"/><span class="lab"><font color="Green" title="E28">A189</font></span>: 
<font color="Maroon" title="c9">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font>) holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">i2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E3:1_1_1_5_1_1_2"/><span class="lab"><font color="Green" title="E29">A190</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">l</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5_1_1_2"><span class="lab"><font color="Green" title="E27">A188</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E4:1_1_1_5_1_1_2"/>
<font color="Maroon" title="c9">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_1"><span class="lab"><font color="Green" title="E24">A183</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_1_2"><span class="lab"><font color="Green" title="E27">A188</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:1_1_1_5_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5_1_1_2"><span class="lab"><font color="Green" title="E29">A190</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font>) holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">i2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_1"><span class="lab"><font color="Green" title="E24">A183</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_1_2"><span class="lab"><font color="Green" title="E28">A189</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:1_1_1_5_1_1"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5_1_1"><span class="lab"><font color="Green" title="E26">A185</font></span></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> holds <br/>(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E6:1_1_1_5_1_1"/><span class="lab"><font color="Green" title="E27">A191</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">and </span><br/><a NAME="E7:1_1_1_5_1_1"/><span class="lab"><font color="Green" title="E28">A192</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><a NAME="E8:1_1_1_5_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E9:1_1_1_5_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_1"><span class="lab"><font color="Green" title="E24">A183</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_1"><span class="lab"><font color="Green" title="E28">A192</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_5_1_1"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_5_1_1"><span class="lab"><font color="Green" title="E27">A191</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E24">A193</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E25">A194</font></span>: 
<span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_1_1"><span class="lab"><font color="Green" title="E18">A177</font></span></a>, <a class="ref" href="jordan5b.html#T1" title="JORDAN5B:th.1">JORDAN5B:1</a></span>;<br/><a NAME="E3:1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E26">A195</font></span>: 
 <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E4:1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E27">A196</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="ref" href="jordan11.html#T8" title="JORDAN11:th.8">JORDAN11:8</a></span>;<br/><a NAME="E5:1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E28">A197</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</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"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E24">A193</font></span></a></span>;<br/><a NAME="E6:1_1_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A198</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</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="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/><a NAME="E7:1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E30">A199</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="ref" href="jordan11.html#T9" title="JORDAN11:th.9">JORDAN11:9</a></span>;<br/><a NAME="E8:1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E31">A200</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E28">A197</font></span></a>, <a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>;<br/><div><span class="lab"><font color="Green" title="E32">A201</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c9">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies  for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font>) holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">i2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_5_1_2_1"/><span class="lab"><font color="Green" title="E33">A202</font></span>: 
<font color="Maroon" title="c9">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_2_1"/><span class="lab"><font color="Green" title="E34">A203</font></span>: 
<font color="Maroon" title="c9">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font>) holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b1">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b3">i2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">let </span><font color="Maroon" title="c10">i1</font>, <font color="Maroon" title="c11">j1</font>, <font color="Maroon" title="c12">i2</font>, <font color="Maroon" title="c13">j2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">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="c12">i2</font>,<font color="Maroon" title="c13">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">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="c10">i1</font>,<font color="Maroon" title="c11">j1</font>) &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c12">i2</font>,<font color="Maroon" title="c13">j2</font>) implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">i2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:1_1_1_5_1_2_1"/><span class="lab"><font color="Green" title="E35">A204</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_2_1"/><span class="lab"><font color="Green" title="E36">A205</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<font color="Maroon" title="c13">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_5_1_2_1"/><span class="lab"><font color="Green" title="E37">A206</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">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="c10">i1</font>,<font color="Maroon" title="c11">j1</font>)
 <span class="kw">and </span><br/><a NAME="E6:1_1_1_5_1_2_1"/><span class="lab"><font color="Green" title="E38">A207</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">l</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c12">i2</font>,<font color="Maroon" title="c13">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">i2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E7:1_1_1_5_1_2_1"/>
<font color="Maroon" title="c9">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E24">A193</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E33">A202</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:1_1_1_5_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A208</font></span>: 
( <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></span>;<br/><a NAME="E9:1_1_1_5_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A209</font></span>: 
<font color="Maroon" title="c12">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E24">A193</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E31">A200</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E30">A199</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E33">A202</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E34">A203</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E36">A205</font></span></a>, <a class="txt" href="#E6:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E38">A207</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E10:1_1_1_5_1_2_1"/><span class="lab"><font color="Green" title="E41">A210</font></span>: 
<font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E29">A198</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E31">A200</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E27">A196</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E30">A199</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E33">A202</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E39">A208</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E35">A204</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E37">A206</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E11:1_1_1_5_1_2_1"/>
<font color="Maroon" title="c13">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E24">A193</font></span></a>, <a class="txt" href="#E6:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E29">A198</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E31">A200</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E27">A196</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E30">A199</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E34">A203</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E39">A208</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E36">A205</font></span></a>, <a class="txt" href="#E6:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E38">A207</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E12:1_1_1_5_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A211</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E41">A210</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E13:1_1_1_5_1_2_1"/>
<font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E24">A193</font></span></a>, <a class="txt" href="#E6:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E29">A198</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E27">A196</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E33">A202</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E34">A203</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E39">A208</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E35">A204</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E37">A206</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E14:1_1_1_5_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1"><span class="lab"><font color="Green" title="E3">A3</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E40">A209</font></span></a>, <a class="ref" href="nat_d.html#T43" title="NAT_D:th.43">NAT_D:43</a>, <a class="ref" href="nat_d.html#T55" title="NAT_D:th.55">NAT_D:55</a></span>;<br/><span class="kw">hence </span><a NAME="E15:1_1_1_5_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="c10">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c12">i2</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c13">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_1_5_1_2_1"><span class="lab"><font color="Green" title="E42">A211</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">l</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) ) )</span><br/><span class="kw">assume </span><a NAME="E1:1_1_1_5_1_2_2"/><span class="lab"><font color="Green" title="E33">A212</font></span>: 
<font color="Maroon" title="c9">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) )</span><br/><a NAME="E2:1_1_1_5_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c9">l</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E24">A193</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:1_1_1_5_1_2_2"/><span class="kw">then </span>
( <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_5_1_2_2"/>
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">l</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E29">A198</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E31">A200</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E27">A196</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E30">A199</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_2_2"><span class="lab"><font color="Green" title="E33">A212</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E33">A213</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E32">A201</font></span></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> holds <br/>(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c9">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> &amp; <font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E12:1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E34">A214</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">and </span><br/><a NAME="E13:1_1_1_5_1_2"/><span class="lab"><font color="Green" title="E35">A215</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><a NAME="E14:1_1_1_5_1_2"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E34">A214</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E15:1_1_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A216</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E24">A193</font></span></a>, <a class="txt" href="#E13:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E35">A215</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E16:1_1_1_5_1_2"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E29">A198</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E31">A200</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E27">A196</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E30">A199</font></span></a>, <a class="txt" href="#E11:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E33">A213</font></span></a>, <a class="txt" href="#E13:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E35">A215</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E25">A194</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E26">A195</font></span></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E17:1_1_1_5_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="ref" href="jordan11.html#T11" title="JORDAN11:th.11">JORDAN11:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E18:1_1_1_5_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E29">A198</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E31">A200</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E27">A196</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E30">A199</font></span></a>, <a class="txt" href="#E11:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E33">A213</font></span></a>, <a class="txt" href="#E13:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E35">A215</font></span></a>, <a class="txt" href="#E15:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E36">A216</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E25">A194</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_2"><span class="lab"><font color="Green" title="E26">A195</font></span></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E19:1_1_1_5_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c9">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="ref" href="jordan11.html#T10" title="JORDAN11:th.10">JORDAN11:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E24">A217</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A218</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E26">A219</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E4:1_1_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A220</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E5:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E28">A221</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">i1</font>, <font color="Maroon" title="c10">j1</font>, <font color="Maroon" title="c11">i2</font>, <font color="Maroon" title="c12">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E7:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E29">A222</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E30">A223</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font>)
 <span class="kw">and </span><br/><a NAME="E9:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E31">A224</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E32">A225</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c11">i2</font>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">and </span><br/><a NAME="E11:1_1_1_5_1_3"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/><a NAME="E12:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E33">A226</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="ref" href="jordan1h.html#T58" title="JORDAN1H:th.58">JORDAN1H:58</a></span>;<br/><a NAME="E13:1_1_1_5_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ;<br/><a NAME="E14:1_1_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A227</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="ref" href="jordan1h.html#T57" title="JORDAN1H:th.57">JORDAN1H:57</a></span>;<br/><a NAME="E15:1_1_1_5_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 ;<br/><a NAME="E16:1_1_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A228</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="ref" href="jordan1h.html#T56" title="JORDAN1H:th.56">JORDAN1H:56</a></span>;<br/><a NAME="E17:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E36">A229</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="ref" href="jordan1h.html#T63" title="JORDAN1H:th.63">JORDAN1H:63</a></span>;<br/><a NAME="E18:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T35" title="NAT_D:th.35">NAT_D:35</a></span>;<br/><a NAME="E19:1_1_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A230</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E20:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E38">A231</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="ref" href="jordan1h.html#T62" title="JORDAN1H:th.62">JORDAN1H:62</a></span>;<br/><a NAME="E21:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E39">A232</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="ref" href="jordan1h.html#T61" title="JORDAN1H:th.61">JORDAN1H:61</a></span>;<br/><a NAME="E22:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E40">A233</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="ref" href="jordan1h.html#T60" title="JORDAN1H:th.60">JORDAN1H:60</a></span>;<br/><a NAME="E23:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E41">A234</font></span>: 
(  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="ref" href="jordan1h.html#T59" title="JORDAN1H:th.59">JORDAN1H:59</a></span>;<br/><a NAME="E24:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E42">A235</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E25:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E43">A236</font></span>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_5"><span class="lab"><font color="Green" title="E21">A180</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a></span>;<br/><a NAME="E26:1_1_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A237</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c9">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="ref" href="jordan1h.html#T52" title="JORDAN1H:th.52">JORDAN1H:52</a></span>;<br/><a NAME="E27:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E45">A238</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E25:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E43">A236</font></span></a>, <a class="ref" href="jordan1h.html#T55" title="JORDAN1H:th.55">JORDAN1H:55</a></span>;<br/><a NAME="E28:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E46">A239</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E25:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E43">A236</font></span></a>, <a class="ref" href="jordan1h.html#T54" title="JORDAN1H:th.54">JORDAN1H:54</a></span>;<br/><a NAME="E29:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E47">A240</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E25:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E43">A236</font></span></a>, <a class="ref" href="jordan1h.html#T53" title="JORDAN1H:th.53">JORDAN1H:53</a></span>;<br/><a NAME="E30:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E48">A241</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><span class="kw">thus </span><a NAME="E31:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E49">A242</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> holds <br/>(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_5_1_3_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_3_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="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> <span class="kw"> such that </span><br/><a NAME="E3:1_1_1_5_1_3_1_1_1"/><span class="lab"><font color="Green" title="E50">A243</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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> <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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_1_1_1"/><span class="lab"><font color="Green" title="E51">A244</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E25:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E43">A236</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c15">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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>;<br/><a NAME="E5:1_1_1_5_1_3_1_1_1"/><span class="lab"><font color="Green" title="E52">A245</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a></span>;<br/><a NAME="E6:1_1_1_5_1_3_1_1_1"/><span class="lab"><font color="Green" title="E53">A246</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c11">i2</font>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E25">A218</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E7:1_1_1_5_1_3_1_1_1"/><span class="lab"><font color="Green" title="E54">A247</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E19:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E37">A230</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><span class="kw">thus </span><a NAME="E8:1_1_1_5_1_3_1_1_1"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_5_1_3_1_1_1_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E27">A220</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_1"><span class="lab"><font color="Green" title="E50">A243</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_1_1_1"><span class="lab"><font color="Green" title="E54">A247</font></span></a>, <a class="txt" href="#E6:1_1_1_5_1_3_1_1_1"><span class="lab"><font color="Green" title="E53">A246</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_1_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E55">A248</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E56">A249</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">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="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_5_1_3_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E57">A250</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E58">A251</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_5_1_3_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E59">A252</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A253</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">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="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E5:1_1_1_5_1_3_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E61">A254</font></span>: 
<font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E44">A237</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E55">A248</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A251</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A253</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E6:1_1_1_5_1_3_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E57">A250</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E59">A252</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E7:1_1_1_5_1_3_1_1_1_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">i19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E48">A241</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E61">A254</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E8:1_1_1_5_1_3_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A255</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i29</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E9:1_1_1_5_1_3_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E63">A256</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E44">A237</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E55">A248</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A251</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A253</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E10:1_1_1_5_1_3_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E57">A250</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E59">A252</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E11:1_1_1_5_1_3_1_1_1_1_1_1_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E63">A256</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E12:1_1_1_5_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="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A255</font></span></a>, <a class="ref" href="uniform1.html#T11" title="UNIFORM1:th.11">UNIFORM1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_5_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E26:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E44">A237</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_1"><span class="lab"><font color="Green" title="E51">A244</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_1"><span class="lab"><font color="Green" title="E52">A245</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E55">A248</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A249</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_1_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E55">A257</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E56">A258</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_5_1_3_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E57">A259</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E58">A260</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_5_1_3_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E59">A261</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E60">A262</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E5:1_1_1_5_1_3_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E61">A263</font></span>: 
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E47">A240</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E55">A257</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E58">A260</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A262</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E6:1_1_1_5_1_3_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A259</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E59">A261</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E7:1_1_1_5_1_3_1_1_1_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A264</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_5_1_3_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E61">A263</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E8:1_1_1_5_1_3_1_1_1_1_1_2_1"/><span class="lab"><font color="Green" title="E63">A265</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E47">A240</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E55">A257</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E58">A260</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E60">A262</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_5_1_3_1_1_1_1_1_2_1"/>
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E57">A259</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E59">A261</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_5_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="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E63">A265</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_1_1_1_1_1_2_1"><span class="lab"><font color="Green" title="E62">A264</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_5_1_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E29:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E47">A240</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_1"><span class="lab"><font color="Green" title="E51">A244</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_1"><span class="lab"><font color="Green" title="E52">A245</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E55">A257</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A258</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_1_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E55">A266</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E56">A267</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_5_1_3_1_1_1_1_1_3_1"/><span class="lab"><font color="Green" title="E57">A268</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_1_1_1_3_1"/><span class="lab"><font color="Green" title="E58">A269</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_5_1_3_1_1_1_1_1_3_1"/><span class="lab"><font color="Green" title="E59">A270</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_1_1_1_1_1_3_1"/><span class="lab"><font color="Green" title="E60">A271</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E5:1_1_1_5_1_3_1_1_1_1_1_3_1"/><span class="lab"><font color="Green" title="E61">A272</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E46">A239</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E55">A266</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A269</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E60">A271</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E6:1_1_1_5_1_3_1_1_1_1_1_3_1"/>
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A268</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A270</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E7:1_1_1_5_1_3_1_1_1_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">j19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E42">A235</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E61">A272</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E8:1_1_1_5_1_3_1_1_1_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A273</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j29</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E9:1_1_1_5_1_3_1_1_1_1_1_3_1"/><span class="lab"><font color="Green" title="E63">A274</font></span>: 
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E46">A239</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E55">A266</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E58">A269</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E60">A271</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E10:1_1_1_5_1_3_1_1_1_1_1_3_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E57">A268</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E59">A270</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E11:1_1_1_5_1_3_1_1_1_1_1_3_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3_1_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E63">A274</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E12:1_1_1_5_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="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3_1_1_1_1_1_3_1"><span class="lab"><font color="Green" title="E62">A273</font></span></a>, <a class="ref" href="uniform1.html#T11" title="UNIFORM1:th.11">UNIFORM1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_5_1_3_1_1_1_1_1_3"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E28:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E46">A239</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_1"><span class="lab"><font color="Green" title="E51">A244</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_1"><span class="lab"><font color="Green" title="E52">A245</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E55">A266</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E56">A267</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_1_1_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E55">A275</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E56">A276</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_1_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">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="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_5_1_3_1_1_1_1_1_4_1"/><span class="lab"><font color="Green" title="E57">A277</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_1_1_1_4_1"/><span class="lab"><font color="Green" title="E58">A278</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_5_1_3_1_1_1_1_1_4_1"/><span class="lab"><font color="Green" title="E59">A279</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_1_1_1_1_1_4_1"/><span class="lab"><font color="Green" title="E60">A280</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">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="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E5:1_1_1_5_1_3_1_1_1_1_1_4_1"/><span class="lab"><font color="Green" title="E61">A281</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E45">A238</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E55">A275</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E58">A278</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E60">A280</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E6:1_1_1_5_1_3_1_1_1_1_1_4_1"/>
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E57">A277</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E59">A279</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E7:1_1_1_5_1_3_1_1_1_1_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A282</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_5_1_3_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E61">A281</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E8:1_1_1_5_1_3_1_1_1_1_1_4_1"/><span class="lab"><font color="Green" title="E63">A283</font></span>: 
<font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E45">A238</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E55">A275</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E58">A278</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E60">A280</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_5_1_3_1_1_1_1_1_4_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E57">A277</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E59">A279</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_5_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="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E63">A283</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_1_1_1_1_1_4_1"><span class="lab"><font color="Green" title="E62">A282</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_5_1_3_1_1_1_1_1_4"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E27:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E45">A238</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_1"><span class="lab"><font color="Green" title="E51">A244</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_1"><span class="lab"><font color="Green" title="E52">A245</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E55">A275</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E56">A276</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_3_1_1_2"/><span class="lab"><font color="Green" title="E50">A284</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="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> <span class="kw"> such that </span><br/><a NAME="E3:1_1_1_5_1_3_1_1_2"/><span class="lab"><font color="Green" title="E51">A285</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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> <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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_1_1_2"/><span class="lab"><font color="Green" title="E52">A286</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E25:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E43">A236</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c15">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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>;<br/><a NAME="E5:1_1_1_5_1_3_1_1_2"/><span class="lab"><font color="Green" title="E53">A287</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a></span>;<br/><a NAME="E6:1_1_1_5_1_3_1_1_2"/><span class="lab"><font color="Green" title="E54">A288</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c11">i2</font>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E25">A218</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E7:1_1_1_5_1_3_1_1_2"/><span class="lab"><font color="Green" title="E55">A289</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E19:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E37">A230</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><span class="kw">thus </span><a NAME="E8:1_1_1_5_1_3_1_1_2"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_5_1_3_1_1_2_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E27">A220</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E51">A285</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E55">A289</font></span></a>, <a class="txt" href="#E6:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E54">A288</font></span></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_2_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E56">A290</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E57">A291</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_2_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_5_1_3_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E58">A292</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E59">A293</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_5_1_3_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A294</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E61">A295</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E5:1_1_1_5_1_3_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A296</font></span>: 
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E58">A292</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E60">A294</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E6:1_1_1_5_1_3_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E35">A228</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E50">A284</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E56">A290</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E59">A293</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E61">A295</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E7:1_1_1_5_1_3_1_1_2_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A297</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_5_1_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E62">A296</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E8:1_1_1_5_1_3_1_1_2_1_1_1_1"/><span class="lab"><font color="Green" title="E64">A298</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E58">A292</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E60">A294</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_5_1_3_1_1_2_1_1_1_1"/>
<font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E35">A228</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E50">A284</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E56">A290</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E59">A293</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E61">A295</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_5_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="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E64">A298</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_1_1_2_1_1_1_1"><span class="lab"><font color="Green" title="E63">A297</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_5_1_3_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E16:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E35">A228</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E50">A284</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E52">A286</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E53">A287</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E56">A290</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E57">A291</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_2_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E56">A299</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E57">A300</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">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="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_5_1_3_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E58">A301</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E59">A302</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_5_1_3_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E60">A303</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E61">A304</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">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="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E5:1_1_1_5_1_3_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E62">A305</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E58">A301</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E60">A303</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E6:1_1_1_5_1_3_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E34">A227</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E50">A284</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A299</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E59">A302</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E61">A304</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E7:1_1_1_5_1_3_1_1_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A306</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_5_1_3_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E62">A305</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E8:1_1_1_5_1_3_1_1_2_1_1_2_1"/><span class="lab"><font color="Green" title="E64">A307</font></span>: 
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E58">A301</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E60">A303</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_5_1_3_1_1_2_1_1_2_1"/>
<font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E34">A227</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E50">A284</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A299</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E59">A302</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E61">A304</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_5_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="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E64">A307</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_1_1_2_1_1_2_1"><span class="lab"><font color="Green" title="E63">A306</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_5_1_3_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E14:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E34">A227</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E50">A284</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E52">A286</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E53">A287</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A299</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E57">A300</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_2_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E56">A308</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E57">A309</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_2_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">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="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_5_1_3_1_1_2_1_1_3_1"/><span class="lab"><font color="Green" title="E58">A310</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_2_1_1_3_1"/><span class="lab"><font color="Green" title="E59">A311</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_5_1_3_1_1_2_1_1_3_1"/><span class="lab"><font color="Green" title="E60">A312</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_1_1_2_1_1_3_1"/><span class="lab"><font color="Green" title="E61">A313</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">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="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E5:1_1_1_5_1_3_1_1_2_1_1_3_1"/><span class="lab"><font color="Green" title="E62">A314</font></span>: 
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E58">A310</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E60">A312</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E6:1_1_1_5_1_3_1_1_2_1_1_3_1"/>
<font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E33">A226</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E50">A284</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E56">A308</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E59">A311</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E61">A313</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E7:1_1_1_5_1_3_1_1_2_1_1_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">i19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E48">A241</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E62">A314</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E8:1_1_1_5_1_3_1_1_2_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A315</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i29</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E9:1_1_1_5_1_3_1_1_2_1_1_3_1"/><span class="lab"><font color="Green" title="E64">A316</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E58">A310</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E60">A312</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E10:1_1_1_5_1_3_1_1_2_1_1_3_1"/>
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E33">A226</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E50">A284</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E56">A308</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E59">A311</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E61">A313</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E11:1_1_1_5_1_3_1_1_2_1_1_3_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3_1_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E64">A316</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E12:1_1_1_5_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="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3_1_1_2_1_1_3_1"><span class="lab"><font color="Green" title="E63">A315</font></span></a>, <a class="ref" href="uniform1.html#T11" title="UNIFORM1:th.11">UNIFORM1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_5_1_3_1_1_2_1_1_3"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E12:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E33">A226</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E50">A284</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E52">A286</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E53">A287</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E56">A308</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E57">A309</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_2_1_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E56">A317</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E57">A318</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_2_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_5_1_3_1_1_2_1_1_4_1"/><span class="lab"><font color="Green" title="E58">A319</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_2_1_1_4_1"/><span class="lab"><font color="Green" title="E59">A320</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_5_1_3_1_1_2_1_1_4_1"/><span class="lab"><font color="Green" title="E60">A321</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_1_1_2_1_1_4_1"/><span class="lab"><font color="Green" title="E61">A322</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E5:1_1_1_5_1_3_1_1_2_1_1_4_1"/><span class="lab"><font color="Green" title="E62">A323</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_4_1"><span class="lab"><font color="Green" title="E58">A319</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_2_1_1_4_1"><span class="lab"><font color="Green" title="E60">A321</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E6:1_1_1_5_1_3_1_1_2_1_1_4_1"/>
<font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E41">A234</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E50">A284</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E56">A317</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_2_1_1_4_1"><span class="lab"><font color="Green" title="E59">A320</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_2_1_1_4_1"><span class="lab"><font color="Green" title="E61">A322</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E7:1_1_1_5_1_3_1_1_2_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">j19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E42">A235</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_2_1_1_4_1"><span class="lab"><font color="Green" title="E62">A323</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E8:1_1_1_5_1_3_1_1_2_1_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A324</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j29</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E9:1_1_1_5_1_3_1_1_2_1_1_4_1"/><span class="lab"><font color="Green" title="E64">A325</font></span>: 
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_4_1"><span class="lab"><font color="Green" title="E58">A319</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_2_1_1_4_1"><span class="lab"><font color="Green" title="E60">A321</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E10:1_1_1_5_1_3_1_1_2_1_1_4_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E41">A234</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E50">A284</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E56">A317</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_2_1_1_4_1"><span class="lab"><font color="Green" title="E59">A320</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_2_1_1_4_1"><span class="lab"><font color="Green" title="E61">A322</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E11:1_1_1_5_1_3_1_1_2_1_1_4_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3_1_1_2_1_1_4_1"><span class="lab"><font color="Green" title="E64">A325</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E12:1_1_1_5_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="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3_1_1_2_1_1_4_1"><span class="lab"><font color="Green" title="E63">A324</font></span></a>, <a class="ref" href="uniform1.html#T11" title="UNIFORM1:th.11">UNIFORM1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_5_1_3_1_1_2_1_1_4"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E23:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E41">A234</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E50">A284</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E52">A286</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_2"><span class="lab"><font color="Green" title="E53">A287</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E56">A317</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E57">A318</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_3_1_1_3"/><span class="lab"><font color="Green" title="E50">A326</font></span>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="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> <span class="kw"> such that </span><br/><a NAME="E3:1_1_1_5_1_3_1_1_3"/><span class="lab"><font color="Green" title="E51">A327</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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> <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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_1_1_3"/><span class="lab"><font color="Green" title="E52">A328</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E25:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E43">A236</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c15">f</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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>;<br/><a NAME="E5:1_1_1_5_1_3_1_1_3"/><span class="lab"><font color="Green" title="E53">A329</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a></span>;<br/><a NAME="E6:1_1_1_5_1_3_1_1_3"/><span class="lab"><font color="Green" title="E54">A330</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c11">i2</font>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E25">A218</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E7:1_1_1_5_1_3_1_1_3"/><span class="lab"><font color="Green" title="E55">A331</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E19:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E37">A230</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><span class="kw">thus </span><a NAME="E8:1_1_1_5_1_3_1_1_3"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_3_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_5_1_3_1_1_3_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E27">A220</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E51">A327</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E55">A331</font></span></a>, <a class="txt" href="#E6:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E54">A330</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_3_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E56">A332</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E57">A333</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_3_1_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">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="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_5_1_3_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E58">A334</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E59">A335</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_5_1_3_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A336</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E61">A337</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">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="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E5:1_1_1_5_1_3_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A338</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E58">A334</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A336</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E6:1_1_1_5_1_3_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E40">A233</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E50">A326</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A332</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E59">A335</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A337</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E7:1_1_1_5_1_3_1_1_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A339</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_5_1_3_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E62">A338</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E8:1_1_1_5_1_3_1_1_3_1_1_1_1"/><span class="lab"><font color="Green" title="E64">A340</font></span>: 
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E58">A334</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E60">A336</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_5_1_3_1_1_3_1_1_1_1"/>
<font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E40">A233</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E50">A326</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A332</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E59">A335</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E61">A337</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_5_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="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E64">A340</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_1_1_3_1_1_1_1"><span class="lab"><font color="Green" title="E63">A339</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_5_1_3_1_1_3_1_1_1"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E22:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E40">A233</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E50">A326</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E52">A328</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E53">A329</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E56">A332</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E57">A333</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_3_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E56">A341</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E57">A342</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_5_1_3_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E58">A343</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E59">A344</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_5_1_3_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E60">A345</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E61">A346</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E5:1_1_1_5_1_3_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E62">A347</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E58">A343</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E60">A345</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E6:1_1_1_5_1_3_1_1_3_1_1_2_1"/>
<font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E39">A232</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E50">A326</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E56">A341</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E59">A344</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E61">A346</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E7:1_1_1_5_1_3_1_1_3_1_1_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c17">j19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E42">A235</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E62">A347</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E8:1_1_1_5_1_3_1_1_3_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A348</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">j19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c19">j29</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E9:1_1_1_5_1_3_1_1_3_1_1_2_1"/><span class="lab"><font color="Green" title="E64">A349</font></span>: 
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E58">A343</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E60">A345</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E10:1_1_1_5_1_3_1_1_3_1_1_2_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E39">A232</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E50">A326</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E56">A341</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E59">A344</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E61">A346</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E11:1_1_1_5_1_3_1_1_3_1_1_2_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E64">A349</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E12:1_1_1_5_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="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3_1_1_3_1_1_2_1"><span class="lab"><font color="Green" title="E63">A348</font></span></a>, <a class="ref" href="uniform1.html#T11" title="UNIFORM1:th.11">UNIFORM1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_5_1_3_1_1_3_1_1_2"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E21:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E39">A232</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E50">A326</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E52">A328</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E53">A329</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E56">A341</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E57">A342</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_3_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E56">A350</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E57">A351</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_3_1_1_3_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_5_1_3_1_1_3_1_1_3_1"/><span class="lab"><font color="Green" title="E58">A352</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_3_1_1_3_1"/><span class="lab"><font color="Green" title="E59">A353</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_5_1_3_1_1_3_1_1_3_1"/><span class="lab"><font color="Green" title="E60">A354</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_1_1_3_1_1_3_1"/><span class="lab"><font color="Green" title="E61">A355</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E5:1_1_1_5_1_3_1_1_3_1_1_3_1"/><span class="lab"><font color="Green" title="E62">A356</font></span>: 
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E58">A352</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E60">A354</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E6:1_1_1_5_1_3_1_1_3_1_1_3_1"/>
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E38">A231</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E50">A326</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E56">A350</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E59">A353</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E61">A355</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E7:1_1_1_5_1_3_1_1_3_1_1_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A357</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_5_1_3_1_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E62">A356</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E8:1_1_1_5_1_3_1_1_3_1_1_3_1"/><span class="lab"><font color="Green" title="E64">A358</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E58">A352</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E60">A354</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E9:1_1_1_5_1_3_1_1_3_1_1_3_1"/>
<font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E38">A231</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E50">A326</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E56">A350</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E59">A353</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E61">A355</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_5_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="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3_1_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E64">A358</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_1_1_3_1_1_3_1"><span class="lab"><font color="Green" title="E63">A357</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_5_1_3_1_1_3_1_1_3"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E20:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E38">A231</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E50">A326</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E52">A328</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E53">A329</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E56">A350</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E57">A351</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_3_1_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E56">A359</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E57">A360</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="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></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_1_1_3_1_1_4_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">i19</font>, <font color="Maroon" title="c17">j19</font>, <font color="Maroon" title="c18">i29</font>, <font color="Maroon" title="c19">j29</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>) &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">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="c18">i29</font>,<font color="Maroon" title="c19">j29</font>) implies <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_5_1_3_1_1_3_1_1_4_1"/><span class="lab"><font color="Green" title="E58">A361</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i19</font>,<font color="Maroon" title="c17">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_1_1_3_1_1_4_1"/><span class="lab"><font color="Green" title="E59">A362</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c18">i29</font>,<font color="Maroon" title="c19">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_5_1_3_1_1_3_1_1_4_1"/><span class="lab"><font color="Green" title="E60">A363</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c16">i19</font>,<font color="Maroon" title="c17">j19</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_1_1_3_1_1_4_1"/><span class="lab"><font color="Green" title="E61">A364</font></span>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">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="c18">i29</font>,<font color="Maroon" title="c19">j29</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><a NAME="E5:1_1_1_5_1_3_1_1_3_1_1_4_1"/><span class="lab"><font color="Green" title="E62">A365</font></span>: 
<font color="Maroon" title="c11">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E58">A361</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E60">A363</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E6:1_1_1_5_1_3_1_1_3_1_1_4_1"/>
<font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E36">A229</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E50">A326</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E56">A359</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E59">A362</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E61">A364</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E7:1_1_1_5_1_3_1_1_3_1_1_4_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">i19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E48">A241</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E62">A365</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/><a NAME="E8:1_1_1_5_1_3_1_1_3_1_1_4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A366</font></span>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">i19</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c18">i29</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><a NAME="E9:1_1_1_5_1_3_1_1_3_1_1_4_1"/><span class="lab"><font color="Green" title="E64">A367</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E58">A361</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E60">A363</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E10:1_1_1_5_1_3_1_1_3_1_1_4_1"/>
<font color="Maroon" title="c12">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E36">A229</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E50">A326</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E56">A359</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E59">A362</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E61">A364</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E11:1_1_1_5_1_3_1_1_3_1_1_4_1"/><span class="kw">then </span>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E64">A367</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></span>;<br/><span class="kw">hence </span><a NAME="E12:1_1_1_5_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="c18">i29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c16">i19</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j29</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c17">j19</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3_1_1_3_1_1_4_1"><span class="lab"><font color="Green" title="E63">A366</font></span></a>, <a class="ref" href="uniform1.html#T11" title="UNIFORM1:th.11">UNIFORM1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_5_1_3_1_1_3_1_1_4"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E17:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E36">A229</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E50">A326</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E52">A328</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_1_1_3"><span class="lab"><font color="Green" title="E53">A329</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E56">A359</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E57">A360</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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

</div><span class="kw">end;</span></div><span class="kw">let </span><font color="Maroon" title="c13">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font> &amp; <font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> implies (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E32:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E50">A368</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font>
 <span class="kw">and </span><br/><a NAME="E33:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E51">A369</font></span>: 
<font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><a NAME="E34:1_1_1_5_1_3"/><span class="lab"><font color="Green" title="E52">A370</font></span>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_5"><span class="lab"><font color="Green" title="E21">A180</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_5_1_3_2_1"/>
( <font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> or <font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_3_2_1_1"/><span class="lab"><font color="Green" title="E53">A371</font></span>: 
<font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:1_1_1_5_1_3_2_1_1"/><span class="lab"><font color="Green" title="E54">A372</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E42">A235</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E3:1_1_1_5_1_3_2_1_1"/><span class="lab"><font color="Green" title="E55">A373</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E48">A241</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><span class="kw">thus </span><a NAME="E4:1_1_1_5_1_3_2_1_1"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_5_1_3_2_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E56">A374</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:1_1_1_5_1_3_2_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A375</font></span>: 
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E25:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E43">A236</font></span></a></span>;<br/><a NAME="E3:1_1_1_5_1_3_2_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A376</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c11">i2</font>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E25">A218</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E4:1_1_1_5_1_3_2_1_1_1_1_1"/><span class="lab"><font color="Green" title="E59">A377</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E19:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E37">A230</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E57">A375</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E27">A220</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E57">A375</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E59">A377</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A376</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E60">A378</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_2_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E61">A379</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:1_1_1_5_1_3_2_1_1_1_1_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A378</font></span></a>, <a class="ref" href="gobrd13.html#T34" title="GOBRD13:th.34">GOBRD13:34</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E26:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E44">A237</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E55">A373</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A374</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A376</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A378</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E61">A379</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:1_1_1_5_1_3_2_1_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E62">A380</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A378</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E6:1_1_1_5_1_3_2_1_1_1_1_1_1_1_1"/>
 <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="c9">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c10">j1</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E25:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E43">A236</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A378</font></span></a>, <a class="ref" href="gobrd13.html#T21" title="GOBRD13:th.21">GOBRD13:21</a></span>;<br/><span class="kw">hence </span><a NAME="E7:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E26:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E44">A237</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E55">A373</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A376</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E60">A378</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E61">A379</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_2_1_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E62">A380</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E60">A381</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_2_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E61">A382</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:1_1_1_5_1_3_2_1_1_1_1_1_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E60">A381</font></span></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E29:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E47">A240</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A374</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A376</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E60">A381</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E61">A382</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:1_1_1_5_1_3_2_1_1_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E62">A383</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E6:1_1_1_5_1_3_2_1_1_1_1_1_1_1_2"/>
 <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="c9">i1</font>,<font color="Maroon" title="c10">j1</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E25:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E43">A236</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E60">A381</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>;<br/><span class="kw">hence </span><a NAME="E7:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E29:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E47">A240</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A376</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E60">A381</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E61">A382</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_2_1_1_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E62">A383</font></span></a>, <a class="ref" href="gobrd13.html#T21" title="GOBRD13:th.21">GOBRD13:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_1_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E60">A384</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_2_1_1_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E61">A385</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:1_1_1_5_1_3_2_1_1_1_1_1_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E60">A384</font></span></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E28:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E46">A239</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E54">A372</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A374</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A376</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E60">A384</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E61">A385</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:1_1_1_5_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="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E25:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E43">A236</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E60">A384</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E28:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E46">A239</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E54">A372</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A376</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E60">A384</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E61">A385</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_1_1_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E60">A386</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_2_1_1_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E61">A387</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:1_1_1_5_1_3_2_1_1_1_1_1_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E60">A386</font></span></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E27:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E45">A238</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E56">A374</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A376</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E60">A386</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E61">A387</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:1_1_1_5_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>,<font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E25:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E43">A236</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E60">A386</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E27:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E45">A238</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A376</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E60">A386</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E61">A387</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:1_1_1_5_1_3_2_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E56">A388</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:1_1_1_5_1_3_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A389</font></span>: 
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E25:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E43">A236</font></span></a></span>;<br/><a NAME="E3:1_1_1_5_1_3_2_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A390</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c11">i2</font>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E25">A218</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E4:1_1_1_5_1_3_2_1_1_1_1_2"/><span class="lab"><font color="Green" title="E59">A391</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E19:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E37">A230</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E57">A389</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E27">A220</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E57">A389</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E59">A391</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A390</font></span></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E60">A392</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_2_1_1_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E61">A393</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E60">A392</font></span></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E16:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E35">A228</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A388</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A390</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E60">A392</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E61">A393</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E60">A392</font></span></a>, <a class="ref" href="gobrd13.html#T34" title="GOBRD13:th.34">GOBRD13:34</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E16:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E35">A228</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A388</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A390</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E60">A392</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E61">A393</font></span></a>, <a class="ref" href="gobrd13.html#T21" title="GOBRD13:th.21">GOBRD13:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E60">A394</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_2_1_1_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E61">A395</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E60">A394</font></span></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E14:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E34">A227</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A388</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A390</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E60">A394</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E61">A395</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E60">A394</font></span></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E14:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E34">A227</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A388</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A390</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E60">A394</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E61">A395</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_2_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E60">A396</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_2_1_1_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E61">A397</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E60">A396</font></span></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E12:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E33">A226</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E55">A373</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A388</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A390</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E60">A396</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E61">A397</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E60">A396</font></span></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E12:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E33">A226</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E55">A373</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A388</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A390</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E60">A396</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E61">A397</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_2_1_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E60">A398</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_2_1_1_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E61">A399</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E60">A398</font></span></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E23:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E41">A234</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E54">A372</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A388</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A390</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E60">A398</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E61">A399</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E60">A398</font></span></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E23:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E41">A234</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E54">A372</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E56">A388</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A390</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E60">A398</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E61">A399</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:1_1_1_5_1_3_2_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_3"/><span class="lab"><font color="Green" title="E56">A400</font></span>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:1_1_1_5_1_3_2_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A401</font></span>: 
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E25:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E43">A236</font></span></a></span>;<br/><a NAME="E3:1_1_1_5_1_3_2_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A402</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c11">i2</font>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E25">A218</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E4:1_1_1_5_1_3_2_1_1_1_1_3"/><span class="lab"><font color="Green" title="E59">A403</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E19:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E37">A230</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E57">A401</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>) ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>) ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E27">A220</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E57">A401</font></span></a>, <a class="txt" href="#E4:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E59">A403</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A402</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_3_1_1_1"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E60">A404</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_2_1_1_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E61">A405</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:1_1_1_5_1_3_2_1_1_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E62">A406</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E60">A404</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E4:1_1_1_5_1_3_2_1_1_1_1_3_1_1_1"/>
 <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="c9">i1</font>,<font color="Maroon" title="c10">j1</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E34:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E52">A370</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E60">A404</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>;<br/><span class="kw">hence </span><a NAME="E5:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E22:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E40">A233</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E56">A400</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A402</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E60">A404</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E61">A405</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E62">A406</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E6:1_1_1_5_1_3_2_1_1_1_1_3_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E60">A404</font></span></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></span>;<br/><span class="kw">hence </span><a NAME="E7:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E22:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E40">A233</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E56">A400</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A402</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E60">A404</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E61">A405</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_3_1_1_2"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E60">A407</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_2_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E61">A408</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:1_1_1_5_1_3_2_1_1_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E62">A409</font></span>: 
<font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E60">A407</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E4:1_1_1_5_1_3_2_1_1_1_1_3_1_1_2"/>
 <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="c9">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E34:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E52">A370</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E60">A407</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>;<br/><span class="kw">hence </span><a NAME="E5:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E21:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E39">A232</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E54">A372</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E56">A400</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A402</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E60">A407</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E61">A408</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E62">A409</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E6:1_1_1_5_1_3_2_1_1_1_1_3_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E60">A407</font></span></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></span>;<br/><span class="kw">hence </span><a NAME="E7:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E21:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E39">A232</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E54">A372</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E56">A400</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A402</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E60">A407</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E61">A408</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_3_1_1_3"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E60">A410</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_2_1_1_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E61">A411</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c11">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:1_1_1_5_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="c11">i2</font>,<font color="Maroon" title="c12">j2</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E34:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E52">A370</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E60">A410</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E20:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E38">A231</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E56">A400</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A402</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E60">A410</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E61">A411</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:1_1_1_5_1_3_2_1_1_1_1_3_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E60">A410</font></span></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E20:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E38">A231</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E56">A400</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A402</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E60">A410</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E61">A411</font></span></a>, <a class="ref" href="gobrd13.html#T21" title="GOBRD13:th.21">GOBRD13:21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_1_1_1_3_1_1_4"><span class="kw">suppose </span></a><span class="kw">that </span><a NAME="E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E60">A412</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_5_1_3_2_1_1_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E61">A413</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E3:1_1_1_5_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>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E34:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E52">A370</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E60">A412</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E17:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E36">A229</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E55">A373</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E56">A400</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A402</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E60">A412</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E61">A413</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font></span><br/><a NAME="E5:1_1_1_5_1_3_2_1_1_1_1_3_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E28">A221</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E26">A219</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E29">A222</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E30">A223</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E32">A225</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E60">A412</font></span></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="txt" href="#E9:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E31">A224</font></span></a>, <a class="txt" href="#E17:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E36">A229</font></span></a>, <a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E53">A371</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1"><span class="lab"><font color="Green" title="E55">A373</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E56">A400</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A402</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E60">A412</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_1_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E61">A413</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E6:1_1_1_5_1_3_2_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_3_2_1_2"/>
<font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:1_1_1_5_1_3_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E51">A369</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><a NAME="E3:1_1_1_5_1_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A414</font></span>: 
<font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E4:1_1_1_5"><span class="lab"><font color="Green" title="E23">A182</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">i1</font>, <font color="Maroon" title="c15">j1</font>, <font color="Maroon" title="c16">i2</font>, <font color="Maroon" title="c17">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:1_1_1_5_1_3_2_1_2"/><span class="lab"><font color="Green" title="E54">A415</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c14">i1</font>,<font color="Maroon" title="c15">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:1_1_1_5_1_3_2_1_2"/><span class="lab"><font color="Green" title="E55">A416</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">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="c14">i1</font>,<font color="Maroon" title="c15">j1</font>)
 <span class="kw">and </span><br/><a NAME="E7:1_1_1_5_1_3_2_1_2"/><span class="lab"><font color="Green" title="E56">A417</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c16">i2</font>,<font color="Maroon" title="c17">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:1_1_1_5_1_3_2_1_2"/><span class="lab"><font color="Green" title="E57">A418</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i2</font>,<font color="Maroon" title="c17">j2</font>)
 <span class="kw">and </span><br/><a NAME="E9:1_1_1_5_1_3_2_1_2"/><span class="lab"><font color="Green" title="E58">A419</font></span>: 
( ( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> ) or ( <font color="Maroon" title="c14">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> ) or ( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> ) or ( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/><a NAME="E10:1_1_1_5_1_3_2_1_2"/><span class="lab"><font color="Green" title="E59">A420</font></span>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_5"><span class="lab"><font color="Green" title="E21">A180</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E53">A414</font></span></a></span>;<br/><div><span class="lab"><font color="Green" title="E60">A421</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_5_1_3_2_1_2_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_3_2_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">i</font>, <font color="Maroon" title="c19">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:1_1_1_5_1_3_2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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>,<font color="Maroon" title="c19">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E61">A422</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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>,<font color="Maroon" title="c19">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E25:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E43">A236</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c20">i</font> = <font color="Maroon" title="c18">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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="Olive" title="b1">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">take </span><font color="Maroon" title="c21">j</font> = <font color="Maroon" title="c19">j</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E5:1_1_1_5_1_3_2_1_2_1_1_1"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_5_1_3_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E61">A422</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_3_2_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">i</font>, <font color="Maroon" title="c19">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:1_1_1_5_1_3_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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>,<font color="Maroon" title="c19">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E61">A423</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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>,<font color="Maroon" title="c19">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E25:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E43">A236</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c20">i</font> = <font color="Maroon" title="c18">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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="Olive" title="b1">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">take </span><font color="Maroon" title="c21">j</font> = <font color="Maroon" title="c19">j</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E5:1_1_1_5_1_3_2_1_2_1_1_2"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_5_1_3_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E61">A423</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_3_2_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">i</font>, <font color="Maroon" title="c19">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:1_1_1_5_1_3_2_1_2_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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>,<font color="Maroon" title="c19">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_5_1_3_2_1_2_1_1_3"/><span class="lab"><font color="Green" title="E61">A424</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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>,<font color="Maroon" title="c19">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E3:1_1_1_5"><span class="lab"><font color="Green" title="E22">A181</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E24">A217</font></span></a>, <a class="txt" href="#E25:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E43">A236</font></span></a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c20">i</font> = <font color="Maroon" title="c18">i</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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="Olive" title="b1">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">take </span><font color="Maroon" title="c21">j</font> = <font color="Maroon" title="c19">j</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E5:1_1_1_5_1_3_2_1_2_1_1_3"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c20">i</font>,<font color="Maroon" title="c21">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_5_1_3_2_1_2_1_1_3"><span class="lab"><font color="Green" title="E61">A424</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><a NAME="E12:1_1_1_5_1_3_2_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E13:1_1_1_5_1_3_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E53">A414</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E14:1_1_1_5_1_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A425</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c16">i2</font>,<font color="Maroon" title="c17">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E57">A418</font></span></a>, <a class="txt" href="#E11:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E60">A421</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E15:1_1_1_5_1_3_2_1_2"/><span class="lab"><font color="Green" title="E62">A426</font></span>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_5"><span class="lab"><font color="Green" title="E21">A180</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E53">A414</font></span></a></span>;<br/><a NAME="E16:1_1_1_5_1_3_2_1_2"/>
<font color="Maroon" title="c13">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E53">A414</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E17:1_1_1_5_1_3_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E18:1_1_1_5_1_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A427</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c13">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="c14">i1</font>,<font color="Maroon" title="c15">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E55">A416</font></span></a>, <a class="txt" href="#E11:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E60">A421</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_2_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_2_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_5_1_3_2_1_2_2_1"/>
( ( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> ) or ( <font color="Maroon" title="c14">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> ) or ( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> ) or ( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E58">A419</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_2_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_3_2_1_2_2_1_1"/><span class="lab"><font color="Green" title="E64">A428</font></span>: 
( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:1_1_1_5_1_3_2_1_2_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A429</font></span>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="c14">i1</font>,<font color="Maroon" title="c15">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E53">A414</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E54">A415</font></span></a>, <a class="txt" href="#E6:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E55">A416</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E56">A417</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E57">A418</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>;<br/><a NAME="E3:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="c14">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c15">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E53">A414</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E54">A415</font></span></a>, <a class="txt" href="#E6:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E55">A416</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E56">A417</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E57">A418</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_2_2_1_1"><span class="lab"><font color="Green" title="E64">A428</font></span></a>, <a class="ref" href="gobrd13.html#T21" title="GOBRD13:th.21">GOBRD13:21</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_5_1_3_2_1_2_2_1_1"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E33:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E51">A369</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E54">A415</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E56">A417</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E59">A420</font></span></a>, <a class="txt" href="#E15:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E62">A426</font></span></a>, <a class="txt" href="#E18:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E63">A427</font></span></a>, <a class="txt" href="#E14:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E61">A425</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_2_2_1_1"><span class="lab"><font color="Green" title="E64">A428</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_2_2_1_1"><span class="lab"><font color="Green" title="E65">A429</font></span></a>, <a class="ref" href="gobrd13.html#T21" title="GOBRD13:th.21">GOBRD13:21</a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_2_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_3_2_1_2_2_1_2"/><span class="lab"><font color="Green" title="E64">A430</font></span>: 
( <font color="Maroon" title="c14">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:1_1_1_5_1_3_2_1_2_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A431</font></span>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="c14">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">j1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E53">A414</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E54">A415</font></span></a>, <a class="txt" href="#E6:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E55">A416</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E56">A417</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E57">A418</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>;<br/><a NAME="E3:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="c14">i1</font>,<font color="Maroon" title="c15">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E53">A414</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E54">A415</font></span></a>, <a class="txt" href="#E6:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E55">A416</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E56">A417</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E57">A418</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_2_2_1_2"><span class="lab"><font color="Green" title="E64">A430</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_5_1_3_2_1_2_2_1_2"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E33:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E51">A369</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E54">A415</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E56">A417</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E59">A420</font></span></a>, <a class="txt" href="#E15:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E62">A426</font></span></a>, <a class="txt" href="#E18:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E63">A427</font></span></a>, <a class="txt" href="#E14:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E61">A425</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_2_2_1_2"><span class="lab"><font color="Green" title="E64">A430</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_2_2_1_2"><span class="lab"><font color="Green" title="E65">A431</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_2_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_3_2_1_2_2_1_3"/><span class="lab"><font color="Green" title="E64">A432</font></span>: 
( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:1_1_1_5_1_3_2_1_2_2_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A433</font></span>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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">i2</font>,<font color="Maroon" title="c17">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E53">A414</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E54">A415</font></span></a>, <a class="txt" href="#E6:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E55">A416</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E56">A417</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E57">A418</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>;<br/><a NAME="E3:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c17">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E53">A414</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E54">A415</font></span></a>, <a class="txt" href="#E6:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E55">A416</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E56">A417</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E57">A418</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_2_2_1_3"><span class="lab"><font color="Green" title="E64">A432</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_5_1_3_2_1_2_2_1_3"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E33:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E51">A369</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E54">A415</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E56">A417</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E59">A420</font></span></a>, <a class="txt" href="#E15:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E62">A426</font></span></a>, <a class="txt" href="#E18:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E63">A427</font></span></a>, <a class="txt" href="#E14:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E61">A425</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_2_2_1_3"><span class="lab"><font color="Green" title="E64">A432</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_2_2_1_3"><span class="lab"><font color="Green" title="E65">A433</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_5_1_3_2_1_2_2_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_5_1_3_2_1_2_2_1_4"/><span class="lab"><font color="Green" title="E64">A434</font></span>: 
( <font color="Maroon" title="c14">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">i2</font> &amp; <font color="Maroon" title="c15">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/><div class="add"><a NAME="E2:1_1_1_5_1_3_2_1_2_2_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A435</font></span>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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="c14">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c17">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E53">A414</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E54">A415</font></span></a>, <a class="txt" href="#E6:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E55">A416</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E56">A417</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E57">A418</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>;<br/><a NAME="E3:1_1_1_5_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" 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">i2</font>,<font color="Maroon" title="c17">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_5"><span class="lab"><font color="Green" title="E20">A179</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E3:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E53">A414</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E54">A415</font></span></a>, <a class="txt" href="#E6:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E55">A416</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E56">A417</font></span></a>, <a class="txt" href="#E8:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E57">A418</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_2_2_1_4"><span class="lab"><font color="Green" title="E64">A434</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_5_1_3_2_1_2_2_1_4"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E49">A242</font></span></a>, <a class="txt" href="#E32:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E50">A368</font></span></a>, <a class="txt" href="#E33:1_1_1_5_1_3"><span class="lab"><font color="Green" title="E51">A369</font></span></a>, <a class="txt" href="#E5:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E54">A415</font></span></a>, <a class="txt" href="#E7:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E56">A417</font></span></a>, <a class="txt" href="#E10:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E59">A420</font></span></a>, <a class="txt" href="#E15:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E62">A426</font></span></a>, <a class="txt" href="#E18:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E63">A427</font></span></a>, <a class="txt" href="#E14:1_1_1_5_1_3_2_1_2"><span class="lab"><font color="Green" title="E61">A425</font></span></a>, <a class="txt" href="#E1:1_1_1_5_1_3_2_1_2_2_1_4"><span class="lab"><font color="Green" title="E64">A434</font></span></a>, <a class="txt" href="#E2:1_1_1_5_1_3_2_1_2_2_1_4"><span class="lab"><font color="Green" title="E65">A435</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E20:1_1_1_5_1_3_2_1_2"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E36:1_1_1_5_1_3"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c13">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E22:1_1_1"/><span class="lab"><font color="Green" title="E20">A436</font></span>: 
<font color="Maroon">S<sub>4</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:1_1_1_6"/><span class="lab"><font color="Green" title="E21">A437</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> holds <br/> for <font color="Olive" title="b2">m</font>, <font color="Olive" title="b3">k</font>, <font color="Olive" title="b4">i</font>, <font color="Olive" title="b5">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">m</font>,<font color="Olive" title="b3">k</font>) &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b4">i</font>,<font color="Olive" title="b5">j</font>) holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b2">m</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b4">i</font></span>)</span></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Olive" title="b3">k</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Olive" title="b5">j</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1"><span class="lab"><font color="Green" title="E9">A155</font></span></a></span>;<br/>
<a NAME="E2:1_1_1_6"/>
 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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> holds <br/> ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font>) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1"><span class="lab"><font color="Green" title="E9">A155</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:1_1_1_6"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_6"><span class="lab"><font color="Green" title="E21">A437</font></span></a>, <a class="ref" href="goboard1.html#D9" title="GOBOARD1:def.9">GOBOARD1:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> holds <br/>(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font> &amp; <font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> implies (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<font color="Maroon" title="c8">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<font color="Maroon" title="c8">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E4:1_1_1_6"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_6"/><span class="lab"><font color="Green" title="E22">A438</font></span>: 
<font color="Maroon" title="c8">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<font color="Maroon" title="c8">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<font color="Maroon" title="c8">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )</span><br/>

<span class="kw">thus </span><a NAME="E6:1_1_1_6"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<font color="Maroon" title="c8">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">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#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span>,<font color="Maroon" title="c8">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1"><span class="lab"><font color="Green" title="E9">A155</font></span></a>, <a class="txt" href="#E5:1_1_1_6"><span class="lab"><font color="Green" title="E22">A438</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E23:1_1_1"/><span class="lab"><font color="Green" title="E21">A439</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E22:1_1_1"><span class="lab"><font color="Green" title="E20">A436</font></span></a>, <a class="txt" href="#E21:1_1_1"><span class="lab"><font color="Green" title="E19">A178</font></span></a>);<br/></span>
<a NAME="E24:1_1_1"/><span class="lab"><font color="Green" title="E22">A440</font></span>: 
 for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j1</font>, <font color="Olive" title="b5">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i1</font>,<font color="Olive" title="b4">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i1</font>,<font color="Olive" title="b4">j1</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">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>) &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> holds <br/>( <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_7"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font>, <font color="Maroon" title="c9">i1</font>, <font color="Maroon" title="c10">i2</font>, <font color="Maroon" title="c11">j1</font>, <font color="Maroon" title="c12">j2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c10">i2</font>,<font color="Maroon" title="c12">j2</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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies ( <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_7"/><span class="lab"><font color="Green" title="E23">A441</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_7"/><span class="lab"><font color="Green" title="E24">A442</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_7"/><span class="lab"><font color="Green" title="E25">A443</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_7"/><span class="lab"><font color="Green" title="E26">A444</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_7"/><span class="lab"><font color="Green" title="E27">A445</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c10">i2</font>,<font color="Maroon" title="c12">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> or  not  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> or ( <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) )</span><br/>

<a NAME="E6:1_1_1_7"/><span class="lab"><font color="Green" title="E28">A446</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/>
<a NAME="E7:1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A447</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_7"><span class="lab"><font color="Green" title="E23">A441</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/>
<a NAME="E8:1_1_1_7"/>
<span 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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T35" title="NAT_D:th.35">NAT_D:35</a></span>;<br/>
<a NAME="E9:1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A448</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_7"><span class="lab"><font color="Green" title="E29">A447</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E10:1_1_1_7"/><span class="lab"><font color="Green" title="E31">A449</font></span>: 
<font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E11:1_1_1_7"/><span class="lab"><font color="Green" title="E32">A450</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a></span>;<br/>
<a NAME="E12:1_1_1_7"/><span class="lab"><font color="Green" title="E33">A451</font></span>: 
<font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E13:1_1_1_7"/><span class="lab"><font color="Green" title="E34">A452</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_7"><span class="lab"><font color="Green" title="E23">A441</font></span></a>, <a class="txt" href="#E6:1_1_1_7"><span class="lab"><font color="Green" title="E28">A446</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E14:1_1_1_7"/><span class="lab"><font color="Green" title="E35">A453</font></span>: 
<font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E15:1_1_1_7"/><span class="lab"><font color="Green" title="E36">A454</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">j2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E16:1_1_1_7"/><span class="lab"><font color="Green" title="E37">A455</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_7"><span class="lab"><font color="Green" title="E23">A441</font></span></a>, <a class="txt" href="#E6:1_1_1_7"><span class="lab"><font color="Green" title="E28">A446</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<a NAME="E17:1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">A456</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E18:1_1_1_7"/><span class="lab"><font color="Green" title="E39">A457</font></span>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E7:1_1_1_7"><span class="lab"><font color="Green" title="E29">A447</font></span></a>, <a class="txt" href="#E16:1_1_1_7"><span class="lab"><font color="Green" title="E37">A455</font></span></a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_7_1"/><span class="lab"><font color="Green" title="E40">A458</font></span>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_7_1"/><span class="lab"><font color="Green" title="E41">A459</font></span>: 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/><span class="kw">consider </span><font color="Maroon" title="c13">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> <span class="kw"> such that </span><br/><a NAME="E4:1_1_1_7_1"/><span class="lab"><font color="Green" title="E42">A460</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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> <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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_7_1"/><span class="lab"><font color="Green" title="E43">A461</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_7"><span class="lab"><font color="Green" title="E23">A441</font></span></a>, <a class="txt" href="#E11:1_1_1_7"><span class="lab"><font color="Green" title="E32">A450</font></span></a>, <a class="txt" href="#E6:1_1_1_7"><span class="lab"><font color="Green" title="E28">A446</font></span></a>, <a class="txt" href="#E18:1_1_1_7"><span class="lab"><font color="Green" title="E39">A457</font></span></a>, <a class="txt" href="#E1:1_1_1_7_1"><span class="lab"><font color="Green" title="E40">A458</font></span></a>, <a class="txt" href="#E2:1_1_1_7_1"><span class="lab"><font color="Green" title="E41">A459</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E6:1_1_1_7_1"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_7_1"><span class="lab"><font color="Green" title="E42">A460</font></span></a>, <a class="txt" href="#E5:1_1_1_7_1"><span class="lab"><font color="Green" title="E43">A461</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/><a NAME="E7:1_1_1_7_1"/><span class="lab"><font color="Green" title="E44">A462</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_7_1"><span class="lab"><font color="Green" title="E43">A461</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a></span>;<br/><a NAME="E8:1_1_1_7_1"/><span class="lab"><font color="Green" title="E45">A463</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c10">i2</font>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_7"><span class="lab"><font color="Green" title="E27">A445</font></span></a>, <a class="txt" href="#E13:1_1_1_7"><span class="lab"><font color="Green" title="E34">A452</font></span></a>, <a class="txt" href="#E5:1_1_1_7_1"><span class="lab"><font color="Green" title="E43">A461</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E9:1_1_1_7_1"/><span class="lab"><font color="Green" title="E46">A464</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_7"><span class="lab"><font color="Green" title="E25">A443</font></span></a>, <a class="txt" href="#E9:1_1_1_7"><span class="lab"><font color="Green" title="E30">A448</font></span></a>, <a class="txt" href="#E5:1_1_1_7_1"><span class="lab"><font color="Green" title="E43">A461</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_7_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_7"><span class="lab"><font color="Green" title="E24">A442</font></span></a>, <a class="txt" href="#E4:1_1_1_7"><span class="lab"><font color="Green" title="E26">A444</font></span></a>, <a class="txt" href="#E16:1_1_1_7"><span class="lab"><font color="Green" title="E37">A455</font></span></a>, <a class="txt" href="#E17:1_1_1_7"><span class="lab"><font color="Green" title="E38">A456</font></span></a>, <a class="txt" href="#E12:1_1_1_7"><span class="lab"><font color="Green" title="E33">A451</font></span></a>, <a class="txt" href="#E15:1_1_1_7"><span class="lab"><font color="Green" title="E36">A454</font></span></a>, <a class="txt" href="#E4:1_1_1_7_1"><span class="lab"><font color="Green" title="E42">A460</font></span></a>, <a class="txt" href="#E5:1_1_1_7_1"><span class="lab"><font color="Green" title="E43">A461</font></span></a>, <a class="txt" href="#E8:1_1_1_7_1"><span class="lab"><font color="Green" title="E45">A463</font></span></a>, <a class="txt" href="#E7:1_1_1_7_1"><span class="lab"><font color="Green" title="E44">A462</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/><span class="kw">thus </span><a NAME="E11:1_1_1_7_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_7"><span class="lab"><font color="Green" title="E24">A442</font></span></a>, <a class="txt" href="#E4:1_1_1_7"><span class="lab"><font color="Green" title="E26">A444</font></span></a>, <a class="txt" href="#E16:1_1_1_7"><span class="lab"><font color="Green" title="E37">A455</font></span></a>, <a class="txt" href="#E17:1_1_1_7"><span class="lab"><font color="Green" title="E38">A456</font></span></a>, <a class="txt" href="#E10:1_1_1_7"><span class="lab"><font color="Green" title="E31">A449</font></span></a>, <a class="txt" href="#E14:1_1_1_7"><span class="lab"><font color="Green" title="E35">A453</font></span></a>, <a class="txt" href="#E4:1_1_1_7_1"><span class="lab"><font color="Green" title="E42">A460</font></span></a>, <a class="txt" href="#E5:1_1_1_7_1"><span class="lab"><font color="Green" title="E43">A461</font></span></a>, <a class="txt" href="#E9:1_1_1_7_1"><span class="lab"><font color="Green" title="E46">A464</font></span></a>, <a class="txt" href="#E8:1_1_1_7_1"><span class="lab"><font color="Green" title="E45">A463</font></span></a>, <a class="txt" href="#E7:1_1_1_7_1"><span class="lab"><font color="Green" title="E44">A462</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/><span class="kw">thus </span><a NAME="E12:1_1_1_7_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_7"><span class="lab"><font color="Green" title="E24">A442</font></span></a>, <a class="txt" href="#E4:1_1_1_7"><span class="lab"><font color="Green" title="E26">A444</font></span></a>, <a class="txt" href="#E16:1_1_1_7"><span class="lab"><font color="Green" title="E37">A455</font></span></a>, <a class="txt" href="#E17:1_1_1_7"><span class="lab"><font color="Green" title="E38">A456</font></span></a>, <a class="txt" href="#E10:1_1_1_7"><span class="lab"><font color="Green" title="E31">A449</font></span></a>, <a class="txt" href="#E14:1_1_1_7"><span class="lab"><font color="Green" title="E35">A453</font></span></a>, <a class="txt" href="#E4:1_1_1_7_1"><span class="lab"><font color="Green" title="E42">A460</font></span></a>, <a class="txt" href="#E5:1_1_1_7_1"><span class="lab"><font color="Green" title="E43">A461</font></span></a>, <a class="txt" href="#E9:1_1_1_7_1"><span class="lab"><font color="Green" title="E46">A464</font></span></a>, <a class="txt" href="#E8:1_1_1_7_1"><span class="lab"><font color="Green" title="E45">A463</font></span></a>, <a class="txt" href="#E7:1_1_1_7_1"><span class="lab"><font color="Green" title="E44">A462</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )</span><br/><span class="kw">thus </span><a NAME="E13:1_1_1_7_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_7"><span class="lab"><font color="Green" title="E24">A442</font></span></a>, <a class="txt" href="#E4:1_1_1_7"><span class="lab"><font color="Green" title="E26">A444</font></span></a>, <a class="txt" href="#E16:1_1_1_7"><span class="lab"><font color="Green" title="E37">A455</font></span></a>, <a class="txt" href="#E17:1_1_1_7"><span class="lab"><font color="Green" title="E38">A456</font></span></a>, <a class="txt" href="#E12:1_1_1_7"><span class="lab"><font color="Green" title="E33">A451</font></span></a>, <a class="txt" href="#E15:1_1_1_7"><span class="lab"><font color="Green" title="E36">A454</font></span></a>, <a class="txt" href="#E4:1_1_1_7_1"><span class="lab"><font color="Green" title="E42">A460</font></span></a>, <a class="txt" href="#E5:1_1_1_7_1"><span class="lab"><font color="Green" title="E43">A461</font></span></a>, <a class="txt" href="#E9:1_1_1_7_1"><span class="lab"><font color="Green" title="E46">A464</font></span></a>, <a class="txt" href="#E8:1_1_1_7_1"><span class="lab"><font color="Green" title="E45">A463</font></span></a>, <a class="txt" href="#E7:1_1_1_7_1"><span class="lab"><font color="Green" title="E44">A462</font></span></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<span class="kw">defpred </span><font color="Maroon">S<sub>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> ] <span class="kw">means </span> for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">m</font>;<br/>
<a NAME="E25:1_1_1"/><span class="lab"><font color="Green" title="E23">A465</font></span>: 
<font color="Maroon">S<sub>5</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_8"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_1_8"/>
<font color="Maroon" title="c8">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">m</font></span><br/>

<a NAME="E2:1_1_1_8"/><span class="kw">then </span>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">m</font>
 
;<br/>
<span class="kw">hence </span><a NAME="E3:1_1_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1"><span class="lab"><font color="Green" title="E9">A155</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">defpred </span><font color="Maroon">S<sub>6</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span> ex <font color="Olive" title="b1">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b1">w</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; $1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 &amp;  ex <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">w</font></span>)</span> &amp; <font color="Olive" title="b2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">w</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">w</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">w</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">w</font></span>)</span></span>)</span> ) );<br/>
<a NAME="E26:1_1_1"/><span class="lab"><font color="Green" title="E24">A466</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a></span>;<br/>
<a NAME="E27:1_1_1"/><span class="lab"><font color="Green" title="E25">A467</font></span>: 
 for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j1</font>, <font color="Olive" title="b5">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i1</font>,<font color="Olive" title="b4">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i1</font>,<font color="Olive" title="b4">j1</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">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>) &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> holds <br/>( <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_9"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font>, <font color="Maroon" title="c9">i1</font>, <font color="Maroon" title="c10">i2</font>, <font color="Maroon" title="c11">j1</font>, <font color="Maroon" title="c12">j2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c10">i2</font>,<font color="Maroon" title="c12">j2</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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies ( <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_9"/><span class="lab"><font color="Green" title="E26">A468</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_9"/><span class="lab"><font color="Green" title="E27">A469</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_9"/><span class="lab"><font color="Green" title="E28">A470</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_9"/><span class="lab"><font color="Green" title="E29">A471</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_9"/><span class="lab"><font color="Green" title="E30">A472</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c10">i2</font>,<font color="Maroon" title="c12">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> or  not  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> or ( <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) )</span><br/>

<a NAME="E6:1_1_1_9"/><span class="lab"><font color="Green" title="E31">A473</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/>
<a NAME="E7:1_1_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A474</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_9"><span class="lab"><font color="Green" title="E26">A468</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/>
<a NAME="E8:1_1_1_9"/>
<span 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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T35" title="NAT_D:th.35">NAT_D:35</a></span>;<br/>
<a NAME="E9:1_1_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A475</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_9"><span class="lab"><font color="Green" title="E32">A474</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E10:1_1_1_9"/><span class="lab"><font color="Green" title="E34">A476</font></span>: 
<font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E11:1_1_1_9"/><span class="lab"><font color="Green" title="E35">A477</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a></span>;<br/>
<a NAME="E12:1_1_1_9"/><span class="lab"><font color="Green" title="E36">A478</font></span>: 
<font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E13:1_1_1_9"/><span class="lab"><font color="Green" title="E37">A479</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_9"><span class="lab"><font color="Green" title="E26">A468</font></span></a>, <a class="txt" href="#E6:1_1_1_9"><span class="lab"><font color="Green" title="E31">A473</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E14:1_1_1_9"/><span class="lab"><font color="Green" title="E38">A480</font></span>: 
<font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E15:1_1_1_9"/><span class="lab"><font color="Green" title="E39">A481</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">j2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E16:1_1_1_9"/><span class="lab"><font color="Green" title="E40">A482</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_9"><span class="lab"><font color="Green" title="E26">A468</font></span></a>, <a class="txt" href="#E6:1_1_1_9"><span class="lab"><font color="Green" title="E31">A473</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<a NAME="E17:1_1_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A483</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E18:1_1_1_9"/><span class="lab"><font color="Green" title="E42">A484</font></span>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E7:1_1_1_9"><span class="lab"><font color="Green" title="E32">A474</font></span></a>, <a class="txt" href="#E16:1_1_1_9"><span class="lab"><font color="Green" title="E40">A482</font></span></a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span>
<div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_9_1"/><span class="lab"><font color="Green" title="E43">A485</font></span>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_9_1"/><span class="lab"><font color="Green" title="E44">A486</font></span>: 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E4:1_1_1_9_1"/><span class="lab"><font color="Green" title="E45">A487</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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> <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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_9_1"/><span class="lab"><font color="Green" title="E46">A488</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_9"><span class="lab"><font color="Green" title="E26">A468</font></span></a>, <a class="txt" href="#E11:1_1_1_9"><span class="lab"><font color="Green" title="E35">A477</font></span></a>, <a class="txt" href="#E6:1_1_1_9"><span class="lab"><font color="Green" title="E31">A473</font></span></a>, <a class="txt" href="#E18:1_1_1_9"><span class="lab"><font color="Green" title="E42">A484</font></span></a>, <a class="txt" href="#E1:1_1_1_9_1"><span class="lab"><font color="Green" title="E43">A485</font></span></a>, <a class="txt" href="#E2:1_1_1_9_1"><span class="lab"><font color="Green" title="E44">A486</font></span></a></span>;<br/><span class="kw">thus </span><a NAME="E6:1_1_1_9_1"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_9_1"><span class="lab"><font color="Green" title="E45">A487</font></span></a>, <a class="txt" href="#E5:1_1_1_9_1"><span class="lab"><font color="Green" title="E46">A488</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/><a NAME="E7:1_1_1_9_1"/><span class="lab"><font color="Green" title="E47">A489</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_9_1"><span class="lab"><font color="Green" title="E46">A488</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a></span>;<br/><a NAME="E8:1_1_1_9_1"/><span class="lab"><font color="Green" title="E48">A490</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c10">i2</font>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_9"><span class="lab"><font color="Green" title="E30">A472</font></span></a>, <a class="txt" href="#E13:1_1_1_9"><span class="lab"><font color="Green" title="E37">A479</font></span></a>, <a class="txt" href="#E5:1_1_1_9_1"><span class="lab"><font color="Green" title="E46">A488</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E9:1_1_1_9_1"/><span class="lab"><font color="Green" title="E49">A491</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_9"><span class="lab"><font color="Green" title="E28">A470</font></span></a>, <a class="txt" href="#E9:1_1_1_9"><span class="lab"><font color="Green" title="E33">A475</font></span></a>, <a class="txt" href="#E5:1_1_1_9_1"><span class="lab"><font color="Green" title="E46">A488</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_9_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_9"><span class="lab"><font color="Green" title="E27">A469</font></span></a>, <a class="txt" href="#E4:1_1_1_9"><span class="lab"><font color="Green" title="E29">A471</font></span></a>, <a class="txt" href="#E16:1_1_1_9"><span class="lab"><font color="Green" title="E40">A482</font></span></a>, <a class="txt" href="#E17:1_1_1_9"><span class="lab"><font color="Green" title="E41">A483</font></span></a>, <a class="txt" href="#E12:1_1_1_9"><span class="lab"><font color="Green" title="E36">A478</font></span></a>, <a class="txt" href="#E15:1_1_1_9"><span class="lab"><font color="Green" title="E39">A481</font></span></a>, <a class="txt" href="#E4:1_1_1_9_1"><span class="lab"><font color="Green" title="E45">A487</font></span></a>, <a class="txt" href="#E5:1_1_1_9_1"><span class="lab"><font color="Green" title="E46">A488</font></span></a>, <a class="txt" href="#E8:1_1_1_9_1"><span class="lab"><font color="Green" title="E48">A490</font></span></a>, <a class="txt" href="#E7:1_1_1_9_1"><span class="lab"><font color="Green" title="E47">A489</font></span></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/><span class="kw">thus </span><a NAME="E11:1_1_1_9_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_9"><span class="lab"><font color="Green" title="E27">A469</font></span></a>, <a class="txt" href="#E4:1_1_1_9"><span class="lab"><font color="Green" title="E29">A471</font></span></a>, <a class="txt" href="#E16:1_1_1_9"><span class="lab"><font color="Green" title="E40">A482</font></span></a>, <a class="txt" href="#E17:1_1_1_9"><span class="lab"><font color="Green" title="E41">A483</font></span></a>, <a class="txt" href="#E10:1_1_1_9"><span class="lab"><font color="Green" title="E34">A476</font></span></a>, <a class="txt" href="#E14:1_1_1_9"><span class="lab"><font color="Green" title="E38">A480</font></span></a>, <a class="txt" href="#E4:1_1_1_9_1"><span class="lab"><font color="Green" title="E45">A487</font></span></a>, <a class="txt" href="#E5:1_1_1_9_1"><span class="lab"><font color="Green" title="E46">A488</font></span></a>, <a class="txt" href="#E9:1_1_1_9_1"><span class="lab"><font color="Green" title="E49">A491</font></span></a>, <a class="txt" href="#E8:1_1_1_9_1"><span class="lab"><font color="Green" title="E48">A490</font></span></a>, <a class="txt" href="#E7:1_1_1_9_1"><span class="lab"><font color="Green" title="E47">A489</font></span></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/><span class="kw">thus </span><a NAME="E12:1_1_1_9_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_9"><span class="lab"><font color="Green" title="E27">A469</font></span></a>, <a class="txt" href="#E4:1_1_1_9"><span class="lab"><font color="Green" title="E29">A471</font></span></a>, <a class="txt" href="#E16:1_1_1_9"><span class="lab"><font color="Green" title="E40">A482</font></span></a>, <a class="txt" href="#E17:1_1_1_9"><span class="lab"><font color="Green" title="E41">A483</font></span></a>, <a class="txt" href="#E10:1_1_1_9"><span class="lab"><font color="Green" title="E34">A476</font></span></a>, <a class="txt" href="#E14:1_1_1_9"><span class="lab"><font color="Green" title="E38">A480</font></span></a>, <a class="txt" href="#E4:1_1_1_9_1"><span class="lab"><font color="Green" title="E45">A487</font></span></a>, <a class="txt" href="#E5:1_1_1_9_1"><span class="lab"><font color="Green" title="E46">A488</font></span></a>, <a class="txt" href="#E9:1_1_1_9_1"><span class="lab"><font color="Green" title="E49">A491</font></span></a>, <a class="txt" href="#E8:1_1_1_9_1"><span class="lab"><font color="Green" title="E48">A490</font></span></a>, <a class="txt" href="#E7:1_1_1_9_1"><span class="lab"><font color="Green" title="E47">A489</font></span></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )</span><br/><span class="kw">thus </span><a NAME="E13:1_1_1_9_1"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_9"><span class="lab"><font color="Green" title="E27">A469</font></span></a>, <a class="txt" href="#E4:1_1_1_9"><span class="lab"><font color="Green" title="E29">A471</font></span></a>, <a class="txt" href="#E16:1_1_1_9"><span class="lab"><font color="Green" title="E40">A482</font></span></a>, <a class="txt" href="#E17:1_1_1_9"><span class="lab"><font color="Green" title="E41">A483</font></span></a>, <a class="txt" href="#E12:1_1_1_9"><span class="lab"><font color="Green" title="E36">A478</font></span></a>, <a class="txt" href="#E15:1_1_1_9"><span class="lab"><font color="Green" title="E39">A481</font></span></a>, <a class="txt" href="#E4:1_1_1_9_1"><span class="lab"><font color="Green" title="E45">A487</font></span></a>, <a class="txt" href="#E5:1_1_1_9_1"><span class="lab"><font color="Green" title="E46">A488</font></span></a>, <a class="txt" href="#E9:1_1_1_9_1"><span class="lab"><font color="Green" title="E49">A491</font></span></a>, <a class="txt" href="#E8:1_1_1_9_1"><span class="lab"><font color="Green" title="E48">A490</font></span></a>, <a class="txt" href="#E7:1_1_1_9_1"><span class="lab"><font color="Green" title="E47">A489</font></span></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>


</div><span class="kw">end;</span></div>
<a NAME="E28:1_1_1"/><span class="lab"><font color="Green" title="E26">A492</font></span>: 
 for <font color="Olive" title="b1">k</font>, <font color="Olive" title="b2">i1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j1</font>, <font color="Olive" title="b5">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i1</font>,<font color="Olive" title="b4">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b2">i1</font>,<font color="Olive" title="b4">j1</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">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>) &amp;  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> holds <br/>( <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<span class="p2">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i2</font>,<span class="p3">(<span class="default"><font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Olive" title="b2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><font color="Olive" title="b3">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b5">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_10"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font>, <font color="Maroon" title="c9">i1</font>, <font color="Maroon" title="c10">i2</font>, <font color="Maroon" title="c11">j1</font>, <font color="Maroon" title="c12">j2</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c10">i2</font>,<font color="Maroon" title="c12">j2</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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies ( <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) )</span><br/>









<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_10"/><span class="lab"><font color="Green" title="E27">A493</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_10"/><span class="lab"><font color="Green" title="E28">A494</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_10"/><span class="lab"><font color="Green" title="E29">A495</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_10"/><span class="lab"><font color="Green" title="E30">A496</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_10"/><span class="lab"><font color="Green" title="E31">A497</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c10">i2</font>,<font color="Maroon" title="c12">j2</font>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> or ( <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) ) )</span><br/>

<a NAME="E6:1_1_1_10"/><span class="lab"><font color="Green" title="E32">A498</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/>
<a NAME="E7:1_1_1_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A499</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_10"><span class="lab"><font color="Green" title="E27">A493</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<a NAME="E8:1_1_1_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A500</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E9:1_1_1_10"/><span class="lab"><font color="Green" title="E35">A501</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a></span>;<br/>
<span class="kw">assume </span><a NAME="E10:1_1_1_10"/><span class="lab"><font color="Green" title="E36">A502</font></span>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/>

<a NAME="E11:1_1_1_10"/><span class="lab"><font color="Green" title="E37">A503</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_10"><span class="lab"><font color="Green" title="E27">A493</font></span></a>, <a class="txt" href="#E6:1_1_1_10"><span class="lab"><font color="Green" title="E32">A498</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/>
<a NAME="E12:1_1_1_10"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E7:1_1_1_10"><span class="lab"><font color="Green" title="E33">A499</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E14:1_1_1_10"/><span class="lab"><font color="Green" title="E38">A504</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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> <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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">and </span><br/><a NAME="E15:1_1_1_10"/><span class="lab"><font color="Green" title="E39">A505</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a>, <a class="txt" href="#E1:1_1_1_10"><span class="lab"><font color="Green" title="E27">A493</font></span></a>, <a class="txt" href="#E9:1_1_1_10"><span class="lab"><font color="Green" title="E35">A501</font></span></a>, <a class="txt" href="#E6:1_1_1_10"><span class="lab"><font color="Green" title="E32">A498</font></span></a>, <a class="txt" href="#E10:1_1_1_10"><span class="lab"><font color="Green" title="E36">A502</font></span></a></span>;<br/>
<a NAME="E16:1_1_1_10"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_10"><span class="lab"><font color="Green" title="E27">A493</font></span></a>, <a class="txt" href="#E6:1_1_1_10"><span class="lab"><font color="Green" title="E32">A498</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E17:1_1_1_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A506</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c10">i2</font>,<font color="Maroon" title="c12">j2</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_10"><span class="lab"><font color="Green" title="E31">A497</font></span></a>, <a class="txt" href="#E15:1_1_1_10"><span class="lab"><font color="Green" title="E39">A505</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/>
<a NAME="E18:1_1_1_10"/>
<span 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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T35" title="NAT_D:th.35">NAT_D:35</a></span>;<br/>
<a NAME="E19:1_1_1_10"/><span class="kw">then </span>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_1_10"><span class="lab"><font color="Green" title="E37">A503</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E20:1_1_1_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A507</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c11">j1</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_10"><span class="lab"><font color="Green" title="E29">A495</font></span></a>, <a class="txt" href="#E15:1_1_1_10"><span class="lab"><font color="Green" title="E39">A505</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/>
<span class="kw">thus </span><a NAME="E21:1_1_1_10"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_10"><span class="lab"><font color="Green" title="E38">A504</font></span></a>, <a class="txt" href="#E15:1_1_1_10"><span class="lab"><font color="Green" title="E39">A505</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/>

<a NAME="E22:1_1_1_10"/><span class="lab"><font color="Green" title="E42">A508</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c13">i</font>,<font color="Maroon" title="c14">j</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:1_1_1_10"><span class="lab"><font color="Green" title="E39">A505</font></span></a>, <a class="ref" href="finseq_4.html#T67" title="FINSEQ_4:th.67">FINSEQ_4:67</a></span>;<br/>
<a NAME="E23:1_1_1_10"/><span class="lab"><font color="Green" title="E43">A509</font></span>: 
<font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c12">j2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E24:1_1_1_10"/><span class="lab"><font color="Green" title="E44">A510</font></span>: 
<font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c10">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E25:1_1_1_10"/><span class="lab"><font color="Green" title="E45">A511</font></span>: 
<font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c11">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E26:1_1_1_10"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">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="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_10"><span class="lab"><font color="Green" title="E28">A494</font></span></a>, <a class="txt" href="#E4:1_1_1_10"><span class="lab"><font color="Green" title="E30">A496</font></span></a>, <a class="txt" href="#E7:1_1_1_10"><span class="lab"><font color="Green" title="E33">A499</font></span></a>, <a class="txt" href="#E8:1_1_1_10"><span class="lab"><font color="Green" title="E34">A500</font></span></a>, <a class="txt" href="#E23:1_1_1_10"><span class="lab"><font color="Green" title="E43">A509</font></span></a>, <a class="txt" href="#E14:1_1_1_10"><span class="lab"><font color="Green" title="E38">A504</font></span></a>, <a class="txt" href="#E15:1_1_1_10"><span class="lab"><font color="Green" title="E39">A505</font></span></a>, <a class="txt" href="#E20:1_1_1_10"><span class="lab"><font color="Green" title="E41">A507</font></span></a>, <a class="txt" href="#E17:1_1_1_10"><span class="lab"><font color="Green" title="E40">A506</font></span></a>, <a class="txt" href="#E22:1_1_1_10"><span class="lab"><font color="Green" title="E42">A508</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/>

<a NAME="E27:1_1_1_10"/><span class="lab"><font color="Green" title="E46">A512</font></span>: 
<font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c9">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:1_1_1_10"/>
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_10"><span class="lab"><font color="Green" title="E28">A494</font></span></a>, <a class="txt" href="#E4:1_1_1_10"><span class="lab"><font color="Green" title="E30">A496</font></span></a>, <a class="txt" href="#E7:1_1_1_10"><span class="lab"><font color="Green" title="E33">A499</font></span></a>, <a class="txt" href="#E8:1_1_1_10"><span class="lab"><font color="Green" title="E34">A500</font></span></a>, <a class="txt" href="#E24:1_1_1_10"><span class="lab"><font color="Green" title="E44">A510</font></span></a>, <a class="txt" href="#E14:1_1_1_10"><span class="lab"><font color="Green" title="E38">A504</font></span></a>, <a class="txt" href="#E15:1_1_1_10"><span class="lab"><font color="Green" title="E39">A505</font></span></a>, <a class="txt" href="#E20:1_1_1_10"><span class="lab"><font color="Green" title="E41">A507</font></span></a>, <a class="txt" href="#E17:1_1_1_10"><span class="lab"><font color="Green" title="E40">A506</font></span></a>, <a class="txt" href="#E22:1_1_1_10"><span class="lab"><font color="Green" title="E42">A508</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E29:1_1_1_10"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_10"><span class="lab"><font color="Green" title="E28">A494</font></span></a>, <a class="txt" href="#E4:1_1_1_10"><span class="lab"><font color="Green" title="E30">A496</font></span></a>, <a class="txt" href="#E7:1_1_1_10"><span class="lab"><font color="Green" title="E33">A499</font></span></a>, <a class="txt" href="#E8:1_1_1_10"><span class="lab"><font color="Green" title="E34">A500</font></span></a>, <a class="txt" href="#E27:1_1_1_10"><span class="lab"><font color="Green" title="E46">A512</font></span></a>, <a class="txt" href="#E24:1_1_1_10"><span class="lab"><font color="Green" title="E44">A510</font></span></a>, <a class="txt" href="#E14:1_1_1_10"><span class="lab"><font color="Green" title="E38">A504</font></span></a>, <a class="txt" href="#E15:1_1_1_10"><span class="lab"><font color="Green" title="E39">A505</font></span></a>, <a class="txt" href="#E20:1_1_1_10"><span class="lab"><font color="Green" title="E41">A507</font></span></a>, <a class="txt" href="#E17:1_1_1_10"><span class="lab"><font color="Green" title="E40">A506</font></span></a>, <a class="txt" href="#E22:1_1_1_10"><span class="lab"><font color="Green" title="E42">A508</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E30:1_1_1_10"/>
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_10"><span class="lab"><font color="Green" title="E28">A494</font></span></a>, <a class="txt" href="#E4:1_1_1_10"><span class="lab"><font color="Green" title="E30">A496</font></span></a>, <a class="txt" href="#E7:1_1_1_10"><span class="lab"><font color="Green" title="E33">A499</font></span></a>, <a class="txt" href="#E8:1_1_1_10"><span class="lab"><font color="Green" title="E34">A500</font></span></a>, <a class="txt" href="#E25:1_1_1_10"><span class="lab"><font color="Green" title="E45">A511</font></span></a>, <a class="txt" href="#E23:1_1_1_10"><span class="lab"><font color="Green" title="E43">A509</font></span></a>, <a class="txt" href="#E14:1_1_1_10"><span class="lab"><font color="Green" title="E38">A504</font></span></a>, <a class="txt" href="#E15:1_1_1_10"><span class="lab"><font color="Green" title="E39">A505</font></span></a>, <a class="txt" href="#E20:1_1_1_10"><span class="lab"><font color="Green" title="E41">A507</font></span></a>, <a class="txt" href="#E17:1_1_1_10"><span class="lab"><font color="Green" title="E40">A506</font></span></a>, <a class="txt" href="#E22:1_1_1_10"><span class="lab"><font color="Green" title="E42">A508</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E29:1_1_1"/><span class="lab"><font color="Green" title="E27">A513</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 holds <br/>( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Olive" title="b1">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_11"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 implies ( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_1_11"/><span class="lab"><font color="Green" title="E28">A514</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) )</span><br/>

<a NAME="E2:1_1_1_11"/><span class="lab"><font color="Green" title="E29">A515</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a></span>;<br/>
<a NAME="E3:1_1_1_11"/><span class="lab"><font color="Green" title="E30">A516</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/>
<a NAME="E4:1_1_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A517</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_11"><span class="lab"><font color="Green" title="E28">A514</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<a NAME="E5:1_1_1_11"/>
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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_11"><span class="lab"><font color="Green" title="E28">A514</font></span></a>, <a class="txt" href="#E3:1_1_1_11"><span class="lab"><font color="Green" title="E30">A516</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/>
<a NAME="E6:1_1_1_11"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">i1</font>, <font color="Olive" title="b2">j1</font>, <font color="Olive" title="b3">i2</font>, <font color="Olive" title="b4">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i1</font>,<font color="Olive" title="b2">j1</font>) &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b3">i2</font>,<font color="Olive" title="b4">j2</font>) &amp; ( ( <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> ) or ( <font color="Olive" title="b1">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> ) or ( <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> ) or ( <font color="Olive" title="b1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">i2</font> &amp; <font color="Olive" title="b2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_11"><span class="lab"><font color="Green" title="E29">A515</font></span></a>, <a class="txt" href="#E4:1_1_1_11"><span class="lab"><font color="Green" title="E31">A517</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:1_1_1_11"/>
( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1"><span class="lab"><font color="Green" title="E22">A440</font></span></a>, <a class="txt" href="#E27:1_1_1"><span class="lab"><font color="Green" title="E25">A467</font></span></a>, <a class="txt" href="#E28:1_1_1"><span class="lab"><font color="Green" title="E26">A492</font></span></a>, <a class="txt" href="#E1:1_1_1_11"><span class="lab"><font color="Green" title="E28">A514</font></span></a>, <a class="txt" href="#E3:1_1_1_11"><span class="lab"><font color="Green" title="E30">A516</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E30:1_1_1"/><span class="lab"><font color="Green" title="E28">A518</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[1,<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 1,<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a></span>;<br/>
<a NAME="E31:1_1_1"/><span class="lab"><font color="Green" title="E29">A519</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   ex <font color="Olive" title="b2">i</font>, <font color="Olive" title="b3">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b2">i</font>,<font color="Olive" title="b3">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="1_1_1_12"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/>

<a NAME="E1:1_1_1_12"/><span class="lab"><font color="Green" title="E30">A520</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a></span>;<br/>
<a NAME="E2:1_1_1_12"/><span class="lab"><font color="Green" title="E31">A521</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_12_1"/>
( <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 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 )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_1"/><span class="lab"><font color="Green" title="E32">A522</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><span class="kw">take </span>
 <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Olive" title="b1">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Olive" title="b1">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span>
 <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">thus </span><a NAME="E2:1_1_1_12_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="ref" href="jordan11.html#T8" title="JORDAN11:th.8">JORDAN11:8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><a NAME="E3:1_1_1_12_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_12_1_1"><span class="lab"><font color="Green" title="E32">A522</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_12_1_1"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1"><span class="lab"><font color="Green" title="E9">A155</font></span></a>, <a class="txt" href="#E26:1_1_1"><span class="lab"><font color="Green" title="E24">A466</font></span></a>, <a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_2"/><span class="lab"><font color="Green" title="E32">A523</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><span class="kw">take </span>
<span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b1">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Olive" title="b1">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">take </span>
 <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><span class="kw">thus </span><a NAME="E2:1_1_1_12_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="ref" href="jordan11.html#T9" title="JORDAN11:th.9">JORDAN11:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span></span><br/><span class="kw">thus </span><a NAME="E3:1_1_1_12_1_2"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:1_1_1"><span class="lab"><font color="Green" title="E24">A466</font></span></a>, <a class="txt" href="#E30:1_1_1"><span class="lab"><font color="Green" title="E28">A518</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_2"><span class="lab"><font color="Green" title="E32">A523</font></span></a>, <a class="ref" href="finseq_1.html#D9" title="FINSEQ_1:def.9">FINSEQ_1:def 9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_3"/><span class="lab"><font color="Green" title="E32">A524</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:1_1_1_12_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A525</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_12"><span class="lab"><font color="Green" title="E31">A521</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E3:1_1_1_12_1_3"/>
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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_12"><span class="lab"><font color="Green" title="E31">A521</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c9">i1</font>, <font color="Maroon" title="c10">j1</font>, <font color="Maroon" title="c11">i2</font>, <font color="Maroon" title="c12">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:1_1_1_12_1_3"/><span class="lab"><font color="Green" title="E34">A526</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:1_1_1_12_1_3"/><span class="lab"><font color="Green" title="E35">A527</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c9">i1</font>,<font color="Maroon" title="c10">j1</font>)
 <span class="kw">and </span><br/><a NAME="E7:1_1_1_12_1_3"/><span class="lab"><font color="Green" title="E36">A528</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E8:1_1_1_12_1_3"/><span class="lab"><font color="Green" title="E37">A529</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c11">i2</font>,<font color="Maroon" title="c12">j2</font>)
 <span class="kw">and </span><br/><a NAME="E9:1_1_1_12_1_3"/><span class="lab"><font color="Green" title="E38">A530</font></span>: 
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_12"><span class="lab"><font color="Green" title="E30">A520</font></span></a>, <a class="txt" href="#E2:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E33">A525</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_12_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_12_1_3_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_3_1_1_1"/><span class="lab"><font color="Green" title="E39">A531</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_12_1_3_1_1_1_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E38">A530</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_3_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E40">A532</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:1_1_1_12_1_3_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A533</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1"><span class="lab"><font color="Green" title="E22">A440</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_1"><span class="lab"><font color="Green" title="E39">A531</font></span></a></span>;<br/><a NAME="E3:1_1_1_12_1_3_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="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1"><span class="lab"><font color="Green" title="E22">A440</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_1"><span class="lab"><font color="Green" title="E39">A531</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E40">A532</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_12_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_12_1_3_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E41">A533</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_3_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E40">A534</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:1_1_1_12_1_3_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A535</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1"><span class="lab"><font color="Green" title="E22">A440</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_1"><span class="lab"><font color="Green" title="E39">A531</font></span></a></span>;<br/><a NAME="E3:1_1_1_12_1_3_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="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1"><span class="lab"><font color="Green" title="E22">A440</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_1"><span class="lab"><font color="Green" title="E39">A531</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E40">A534</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_12_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_12_1_3_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E41">A535</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_3_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E40">A536</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:1_1_1_12_1_3_1_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A537</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1"><span class="lab"><font color="Green" title="E22">A440</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_1"><span class="lab"><font color="Green" title="E39">A531</font></span></a></span>;<br/><a NAME="E3:1_1_1_12_1_3_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="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1"><span class="lab"><font color="Green" title="E22">A440</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_1"><span class="lab"><font color="Green" title="E39">A531</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E40">A536</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_12_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_12_1_3_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E41">A537</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_3_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E40">A538</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:1_1_1_12_1_3_1_1_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A539</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1"><span class="lab"><font color="Green" title="E22">A440</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_1"><span class="lab"><font color="Green" title="E39">A531</font></span></a></span>;<br/><a NAME="E3:1_1_1_12_1_3_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="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1"><span class="lab"><font color="Green" title="E22">A440</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_1"><span class="lab"><font color="Green" title="E39">A531</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E40">A538</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_12_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_12_1_3_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E41">A539</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_12_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_3_1_1_2"/><span class="lab"><font color="Green" title="E39">A540</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_12_1_3_1_1_2_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E38">A530</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_3_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E40">A541</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:1_1_1_12_1_3_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A542</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1"><span class="lab"><font color="Green" title="E25">A467</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_2"><span class="lab"><font color="Green" title="E39">A540</font></span></a></span>;<br/><a NAME="E3:1_1_1_12_1_3_1_1_2_1_1_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1"><span class="lab"><font color="Green" title="E25">A467</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_2"><span class="lab"><font color="Green" title="E39">A540</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E40">A541</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_12_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_12_1_3_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E41">A542</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_3_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E40">A543</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:1_1_1_12_1_3_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A544</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1"><span class="lab"><font color="Green" title="E25">A467</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_2"><span class="lab"><font color="Green" title="E39">A540</font></span></a></span>;<br/><a NAME="E3:1_1_1_12_1_3_1_1_2_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="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1"><span class="lab"><font color="Green" title="E25">A467</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_2"><span class="lab"><font color="Green" title="E39">A540</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E40">A543</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_12_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_12_1_3_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E41">A544</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_3_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E40">A545</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:1_1_1_12_1_3_1_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A546</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1"><span class="lab"><font color="Green" title="E25">A467</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_2"><span class="lab"><font color="Green" title="E39">A540</font></span></a></span>;<br/><a NAME="E3:1_1_1_12_1_3_1_1_2_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="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1"><span class="lab"><font color="Green" title="E25">A467</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_2"><span class="lab"><font color="Green" title="E39">A540</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E40">A545</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_12_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_12_1_3_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E41">A546</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_2_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_3_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E40">A547</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:1_1_1_12_1_3_1_1_2_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A548</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1"><span class="lab"><font color="Green" title="E25">A467</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_2"><span class="lab"><font color="Green" title="E39">A540</font></span></a></span>;<br/><a NAME="E3:1_1_1_12_1_3_1_1_2_1_1_4"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1"><span class="lab"><font color="Green" title="E25">A467</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_2"><span class="lab"><font color="Green" title="E39">A540</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E40">A547</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_12_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_12_1_3_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E41">A548</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_12_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_3_1_1_3"/><span class="lab"><font color="Green" title="E39">A549</font></span>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_12_1_3_1_1_3_1_1"/>
( ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> ) or ( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E38">A530</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_3_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E40">A550</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:1_1_1_12_1_3_1_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A551</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1"><span class="lab"><font color="Green" title="E26">A492</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_3"><span class="lab"><font color="Green" title="E39">A549</font></span></a></span>;<br/><a NAME="E3:1_1_1_12_1_3_1_1_3_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="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1"><span class="lab"><font color="Green" title="E26">A492</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_3"><span class="lab"><font color="Green" title="E39">A549</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E40">A550</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_12_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_12_1_3_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E41">A551</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_3_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E40">A552</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:1_1_1_12_1_3_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A553</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1"><span class="lab"><font color="Green" title="E26">A492</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_3"><span class="lab"><font color="Green" title="E39">A549</font></span></a></span>;<br/><a NAME="E3:1_1_1_12_1_3_1_1_3_1_1_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1"><span class="lab"><font color="Green" title="E26">A492</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_3"><span class="lab"><font color="Green" title="E39">A549</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E40">A552</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_12_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_12_1_3_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E41">A553</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_3_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E40">A554</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:1_1_1_12_1_3_1_1_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A555</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1"><span class="lab"><font color="Green" title="E26">A492</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_3"><span class="lab"><font color="Green" title="E39">A549</font></span></a></span>;<br/><a NAME="E3:1_1_1_12_1_3_1_1_3_1_1_3"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c11">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1"><span class="lab"><font color="Green" title="E26">A492</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_3"><span class="lab"><font color="Green" title="E39">A549</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E40">A554</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_12_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_12_1_3_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E41">A555</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_3_1_1_3_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_12_1_3_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E40">A556</font></span>: 
( <font color="Maroon" title="c9">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">i2</font> &amp; <font color="Maroon" title="c10">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )</span><br/><div class="add"><a NAME="E2:1_1_1_12_1_3_1_1_3_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A557</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1"><span class="lab"><font color="Green" title="E26">A492</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_3"><span class="lab"><font color="Green" title="E39">A549</font></span></a></span>;<br/><a NAME="E3:1_1_1_12_1_3_1_1_3_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="c11">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c12">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1"><span class="lab"><font color="Green" title="E26">A492</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E32">A524</font></span></a>, <a class="txt" href="#E5:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E34">A526</font></span></a>, <a class="txt" href="#E6:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E35">A527</font></span></a>, <a class="txt" href="#E7:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E36">A528</font></span></a>, <a class="txt" href="#E8:1_1_1_12_1_3"><span class="lab"><font color="Green" title="E37">A529</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_3"><span class="lab"><font color="Green" title="E39">A549</font></span></a>, <a class="txt" href="#E1:1_1_1_12_1_3_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E40">A556</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_12_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_12_1_3_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E41">A557</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_12_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E11:1_1_1_12_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E32:1_1_1"/><span class="lab"><font color="Green" title="E30">A558</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>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#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_13"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c8">k</font>] implies <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_1_13"/><span class="lab"><font color="Green" title="E31">A559</font></span>: 
 for <font color="Olive" title="b1">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c9">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:1_1_1_13"/><span class="lab"><font color="Green" title="E32">A560</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_13_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_13_1"/>
( <font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_13"><span class="lab"><font color="Green" title="E32">A560</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_13_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_13_1_1"/>
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span><br/><div class="add"><a NAME="E2:1_1_1_13_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A561</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E3:1_1_1_13_1_1"/><span class="lab"><font color="Green" title="E34">A562</font></span>: 
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_1"><span class="lab"><font color="Green" title="E29">A519</font></span></a></span>;<br/><a NAME="E4:1_1_1_13_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/><a NAME="E5:1_1_1_13_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_13_1_1"><span class="lab"><font color="Green" title="E33">A561</font></span></a>, <a class="txt" href="#E3:1_1_1_13_1_1"><span class="lab"><font color="Green" title="E34">A562</font></span></a>, <a class="ref" href="finseq_5.html#T22" title="FINSEQ_5:th.22">FINSEQ_5:22</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_13_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_13"><span class="lab"><font color="Green" title="E31">A559</font></span></a>, <a class="txt" href="#E2:1_1_1_13_1_1"><span class="lab"><font color="Green" title="E33">A561</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_13_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_13_1_2"/><span class="lab"><font color="Green" title="E33">A563</font></span>: 
<font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font></span><br/><div class="add"><a NAME="E2:1_1_1_13_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:1_1_1_13_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c9">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_13_1_2"><span class="lab"><font color="Green" title="E33">A563</font></span></a>, <a class="ref" href="finseq_1.html#T58" title="FINSEQ_1:th.58">FINSEQ_1:58</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E33:1_1_1"/><span class="lab"><font color="Green" title="E31">A564</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E25:1_1_1"><span class="lab"><font color="Green" title="E23">A465</font></span></a>, <a class="txt" href="#E32:1_1_1"><span class="lab"><font color="Green" title="E30">A558</font></span></a>);<br/></span>
<a NAME="E34:1_1_1"/><span class="lab"><font color="Green" title="E32">A565</font></span>: 
 for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_14"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">j</font>, <font color="Maroon" title="c9">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">j</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_14"/><span class="lab"><font color="Green" title="E33">A566</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_14"/><span class="lab"><font color="Green" title="E34">A567</font></span>: 
<font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">j</font></span><br/>

<a NAME="E3:1_1_1_14"/>
<font color="Maroon" title="c8">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a>, <a class="txt" href="#E2:1_1_1_14"><span class="lab"><font color="Green" title="E34">A567</font></span></a></span>;<br/>
<a NAME="E4:1_1_1_14"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T59" title="FINSEQ_1:th.59">FINSEQ_1:59</a></span>;<br/>
<a NAME="E5:1_1_1_14"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A568</font></span>: 
<font color="Maroon" title="c8">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_14"><span class="lab"><font color="Green" title="E33">A566</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E6:1_1_1_14"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_1"><span class="lab"><font color="Green" title="E31">A564</font></span></a>, <a class="txt" href="#E2:1_1_1_14"><span class="lab"><font color="Green" title="E34">A567</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:1_1_1_14"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c8">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_14"><span class="lab"><font color="Green" title="E35">A568</font></span></a>, <a class="ref" href="finseq_4.html#T70" title="FINSEQ_4:th.70">FINSEQ_4:70</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">defpred </span><font color="Maroon">S<sub>7</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span><font color="Maroon" title="c7">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="E35:1_1_1"/><span class="lab"><font color="Green" title="E33">A569</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_15"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font>] implies <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_1_15"/><span class="lab"><font color="Green" title="E34">A570</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> is  <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<a NAME="E2:1_1_1_15"/><span class="lab"><font color="Green" title="E35">A571</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_15_1"/>
( <font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 or <font color="Maroon" title="c8">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="1_1_1_15_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_15_1_1"/>
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_15_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E3:1_1_1_15_1_1"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_15_1_1"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sppol_2.html#T26" title="SPPOL_2:th.26">SPPOL_2:26</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E36">A572</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c9">m</font> = <font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1;<br/><a NAME="E2:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E37">A573</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E3:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E38">A574</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/><a NAME="E4:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E39">A575</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="ref" href="nat_d.html#T49" title="NAT_D:th.49">NAT_D:49</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c10">i1</font>, <font color="Maroon" title="c11">j1</font>, <font color="Maroon" title="c12">i2</font>, <font color="Maroon" title="c13">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E6:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E40">A576</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E41">A577</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">j1</font>)
 <span class="kw">and </span><br/><a NAME="E8:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E42">A578</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<font color="Maroon" title="c13">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E9:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E43">A579</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c13">j2</font>)
 <span class="kw">and </span><br/><a NAME="E10:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E44">A580</font></span>: 
( ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_15"><span class="lab"><font color="Green" title="E35">A571</font></span></a>, <a class="txt" href="#E2:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E37">A573</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/><a NAME="E11:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E45">A581</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c10">i1</font>,<font color="Maroon" title="c11">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="c12">i2</font>,<font color="Maroon" title="c13">j2</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E39">A575</font></span></a>, <a class="txt" href="#E2:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E37">A573</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/><a NAME="E12:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E46">A582</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E13:1_1_1_15_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A583</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E14:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E48">A584</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E15:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E49">A585</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E16:1_1_1_15_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A586</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E17:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E51">A587</font></span>: 
<font color="Maroon" title="c10">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E18:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E52">A588</font></span>: 
<font color="Maroon" title="c13">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E19:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E53">A589</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E20:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E54">A590</font></span>: 
<font color="Maroon" title="c11">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E21:1_1_1_15_1_2"/><span class="lab"><font color="Green" title="E55">A591</font></span>: 
<font color="Maroon" title="c12">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_15_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_15_1_2_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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_15_1_2_1_1_1"/><span class="lab"><font color="Green" title="E56">A592</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_15_1_2_1_1_1_1_1"/>
( ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E44">A580</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_15_1_2_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E57">A593</font></span>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_15_1_2_1_1_1_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1"><span class="lab"><font color="Green" title="E22">A440</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_1"><span class="lab"><font color="Green" title="E56">A592</font></span></a></span>;<br/><a NAME="E3:1_1_1_15_1_2_1_1_1_1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E4:1_1_1_15_1_2_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A594</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E17:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E51">A587</font></span></a>, <a class="txt" href="#E14:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E48">A584</font></span></a>, <a class="txt" href="#E18:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E52">A588</font></span></a>, <a class="txt" href="#E16:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E50">A586</font></span></a>, <a class="txt" href="#E11:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E45">A581</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E57">A593</font></span></a>, <a class="ref" href="goboard7.html#T16" title="GOBOARD7:th.16">GOBOARD7:16</a></span>;<br/><a NAME="E5:1_1_1_15_1_2_1_1_1_1_1_1"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1"><span class="lab"><font color="Green" title="E22">A440</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_1"><span class="lab"><font color="Green" title="E56">A592</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E57">A593</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_15_1_2_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_15"><span class="lab"><font color="Green" title="E34">A570</font></span></a>, <a class="txt" href="#E2:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E37">A573</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E4:1_1_1_15_1_2_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E58">A594</font></span></a>, <a class="ref" href="sppol_2.html#T30" title="SPPOL_2:th.30">SPPOL_2:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_15_1_2_1_1_1_1_1_2"/><span class="lab"><font color="Green" title="E57">A595</font></span>: 
( <font color="Maroon" title="c10">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_15_1_2_1_1_1_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1"><span class="lab"><font color="Green" title="E22">A440</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_1"><span class="lab"><font color="Green" title="E56">A592</font></span></a></span>;<br/><a NAME="E3:1_1_1_15_1_2_1_1_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E4:1_1_1_15_1_2_1_1_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A596</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E19:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E53">A589</font></span></a>, <a class="txt" href="#E14:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E48">A584</font></span></a>, <a class="txt" href="#E21:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E55">A591</font></span></a>, <a class="txt" href="#E11:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E45">A581</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E57">A595</font></span></a>, <a class="ref" href="goboard7.html#T18" title="GOBOARD7:th.18">GOBOARD7:18</a></span>;<br/><a NAME="E5:1_1_1_15_1_2_1_1_1_1_1_2"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1"><span class="lab"><font color="Green" title="E22">A440</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_1"><span class="lab"><font color="Green" title="E56">A592</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E57">A595</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_15_1_2_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_15"><span class="lab"><font color="Green" title="E34">A570</font></span></a>, <a class="txt" href="#E2:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E37">A573</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E4:1_1_1_15_1_2_1_1_1_1_1_2"><span class="lab"><font color="Green" title="E58">A596</font></span></a>, <a class="ref" href="sppol_2.html#T30" title="SPPOL_2:th.30">SPPOL_2:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_15_1_2_1_1_1_1_1_3"/><span class="lab"><font color="Green" title="E57">A597</font></span>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_15_1_2_1_1_1_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1"><span class="lab"><font color="Green" title="E22">A440</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_1"><span class="lab"><font color="Green" title="E56">A592</font></span></a></span>;<br/><a NAME="E3:1_1_1_15_1_2_1_1_1_1_1_3"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E4:1_1_1_15_1_2_1_1_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A598</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E17:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E51">A587</font></span></a>, <a class="txt" href="#E20:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E54">A590</font></span></a>, <a class="txt" href="#E15:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E49">A585</font></span></a>, <a class="txt" href="#E13:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E47">A583</font></span></a>, <a class="txt" href="#E11:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E45">A581</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E57">A597</font></span></a>, <a class="ref" href="goboard7.html#T15" title="GOBOARD7:th.15">GOBOARD7:15</a></span>;<br/><a NAME="E5:1_1_1_15_1_2_1_1_1_1_1_3"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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="c13">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1"><span class="lab"><font color="Green" title="E22">A440</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_1"><span class="lab"><font color="Green" title="E56">A592</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E57">A597</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_15_1_2_1_1_1_1_1_3"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_15"><span class="lab"><font color="Green" title="E34">A570</font></span></a>, <a class="txt" href="#E2:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E37">A573</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E4:1_1_1_15_1_2_1_1_1_1_1_3"><span class="lab"><font color="Green" title="E58">A598</font></span></a>, <a class="ref" href="sppol_2.html#T30" title="SPPOL_2:th.30">SPPOL_2:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_15_1_2_1_1_1_1_1_4"/><span class="lab"><font color="Green" title="E57">A599</font></span>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_15_1_2_1_1_1_1_1_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1"><span class="lab"><font color="Green" title="E22">A440</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_1"><span class="lab"><font color="Green" title="E56">A592</font></span></a></span>;<br/><a NAME="E3:1_1_1_15_1_2_1_1_1_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E4:1_1_1_15_1_2_1_1_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A600</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E19:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E53">A589</font></span></a>, <a class="txt" href="#E20:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E54">A590</font></span></a>, <a class="txt" href="#E12:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E46">A582</font></span></a>, <a class="txt" href="#E11:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E45">A581</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E57">A599</font></span></a>, <a class="ref" href="goboard7.html#T17" title="GOBOARD7:th.17">GOBOARD7:17</a></span>;<br/><a NAME="E5:1_1_1_15_1_2_1_1_1_1_1_4"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1"><span class="lab"><font color="Green" title="E22">A440</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_1"><span class="lab"><font color="Green" title="E56">A592</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E57">A599</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_15_1_2_1_1_1_1_1_4"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_15"><span class="lab"><font color="Green" title="E34">A570</font></span></a>, <a class="txt" href="#E2:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E37">A573</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E4:1_1_1_15_1_2_1_1_1_1_1_4"><span class="lab"><font color="Green" title="E58">A600</font></span></a>, <a class="ref" href="sppol_2.html#T30" title="SPPOL_2:th.30">SPPOL_2:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_15_1_2_1_1_1"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_15_1_2_1_1_2"/><span class="lab"><font color="Green" title="E56">A601</font></span>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_15_1_2_1_1_2_1_1"/>
( ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E44">A580</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_15_1_2_1_1_2_1_1_1"/><span class="lab"><font color="Green" title="E57">A602</font></span>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_15_1_2_1_1_2_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1"><span class="lab"><font color="Green" title="E25">A467</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A601</font></span></a></span>;<br/><a NAME="E3:1_1_1_15_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A603</font></span>: 
<font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E4:1_1_1_15_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_15_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E57">A602</font></span></a></span>;<br/><a NAME="E5:1_1_1_15_1_2_1_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A604</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E19:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E53">A589</font></span></a>, <a class="txt" href="#E17:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E51">A587</font></span></a>, <a class="txt" href="#E14:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E48">A584</font></span></a>, <a class="txt" href="#E11:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E45">A581</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E57">A602</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E58">A603</font></span></a>, <a class="ref" href="goboard7.html#T13" title="GOBOARD7:th.13">GOBOARD7:13</a></span>;<br/><a NAME="E6:1_1_1_15_1_2_1_1_2_1_1_1"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1"><span class="lab"><font color="Green" title="E25">A467</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A601</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E57">A602</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:1_1_1_15_1_2_1_1_2_1_1_1"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_15"><span class="lab"><font color="Green" title="E34">A570</font></span></a>, <a class="txt" href="#E2:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E37">A573</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E5:1_1_1_15_1_2_1_1_2_1_1_1"><span class="lab"><font color="Green" title="E59">A604</font></span></a>, <a class="ref" href="sppol_2.html#T30" title="SPPOL_2:th.30">SPPOL_2:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_15_1_2_1_1_2_1_1_2"/><span class="lab"><font color="Green" title="E57">A605</font></span>: 
( <font color="Maroon" title="c10">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_15_1_2_1_1_2_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1"><span class="lab"><font color="Green" title="E25">A467</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A601</font></span></a></span>;<br/><a NAME="E3:1_1_1_15_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A606</font></span>: 
<font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E4:1_1_1_15_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_15_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E57">A605</font></span></a></span>;<br/><a NAME="E5:1_1_1_15_1_2_1_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A607</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E19:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E53">A589</font></span></a>, <a class="txt" href="#E14:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E48">A584</font></span></a>, <a class="txt" href="#E20:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E54">A590</font></span></a>, <a class="txt" href="#E11:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E45">A581</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E57">A605</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E58">A606</font></span></a>, <a class="ref" href="goboard7.html#T14" title="GOBOARD7:th.14">GOBOARD7:14</a></span>;<br/><a NAME="E6:1_1_1_15_1_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1"><span class="lab"><font color="Green" title="E25">A467</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A601</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E57">A605</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:1_1_1_15_1_2_1_1_2_1_1_2"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_15"><span class="lab"><font color="Green" title="E34">A570</font></span></a>, <a class="txt" href="#E2:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E37">A573</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E5:1_1_1_15_1_2_1_1_2_1_1_2"><span class="lab"><font color="Green" title="E59">A607</font></span></a>, <a class="ref" href="sppol_2.html#T30" title="SPPOL_2:th.30">SPPOL_2:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_15_1_2_1_1_2_1_1_3"/><span class="lab"><font color="Green" title="E57">A608</font></span>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_15_1_2_1_1_2_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1"><span class="lab"><font color="Green" title="E25">A467</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A601</font></span></a></span>;<br/><a NAME="E3:1_1_1_15_1_2_1_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A609</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E4:1_1_1_15_1_2_1_1_2_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E5:1_1_1_15_1_2_1_1_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A610</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E17:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E51">A587</font></span></a>, <a class="txt" href="#E14:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E48">A584</font></span></a>, <a class="txt" href="#E20:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E54">A590</font></span></a>, <a class="txt" href="#E16:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E50">A586</font></span></a>, <a class="txt" href="#E11:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E45">A581</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E57">A608</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E58">A609</font></span></a>, <a class="ref" href="goboard7.html#T14" title="GOBOARD7:th.14">GOBOARD7:14</a></span>;<br/><a NAME="E6:1_1_1_15_1_2_1_1_2_1_1_3"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1"><span class="lab"><font color="Green" title="E25">A467</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A601</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E57">A608</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:1_1_1_15_1_2_1_1_2_1_1_3"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_15"><span class="lab"><font color="Green" title="E34">A570</font></span></a>, <a class="txt" href="#E2:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E37">A573</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E5:1_1_1_15_1_2_1_1_2_1_1_3"><span class="lab"><font color="Green" title="E59">A610</font></span></a>, <a class="ref" href="sppol_2.html#T30" title="SPPOL_2:th.30">SPPOL_2:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_2_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_15_1_2_1_1_2_1_1_4"/><span class="lab"><font color="Green" title="E57">A611</font></span>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_15_1_2_1_1_2_1_1_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1"><span class="lab"><font color="Green" title="E25">A467</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A601</font></span></a></span>;<br/><a NAME="E3:1_1_1_15_1_2_1_1_2_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A612</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E4:1_1_1_15_1_2_1_1_2_1_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><a NAME="E5:1_1_1_15_1_2_1_1_2_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A613</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E19:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E53">A589</font></span></a>, <a class="txt" href="#E17:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E51">A587</font></span></a>, <a class="txt" href="#E20:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E54">A590</font></span></a>, <a class="txt" href="#E13:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E47">A583</font></span></a>, <a class="txt" href="#E11:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E45">A581</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E57">A611</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E58">A612</font></span></a>, <a class="ref" href="goboard7.html#T13" title="GOBOARD7:th.13">GOBOARD7:13</a></span>;<br/><a NAME="E6:1_1_1_15_1_2_1_1_2_1_1_4"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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="c13">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1"><span class="lab"><font color="Green" title="E25">A467</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_2"><span class="lab"><font color="Green" title="E56">A601</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E57">A611</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:1_1_1_15_1_2_1_1_2_1_1_4"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_15"><span class="lab"><font color="Green" title="E34">A570</font></span></a>, <a class="txt" href="#E2:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E37">A573</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E5:1_1_1_15_1_2_1_1_2_1_1_4"><span class="lab"><font color="Green" title="E59">A613</font></span></a>, <a class="ref" href="sppol_2.html#T30" title="SPPOL_2:th.30">SPPOL_2:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_15_1_2_1_1_2"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_15_1_2_1_1_3"/><span class="lab"><font color="Green" title="E56">A614</font></span>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_15_1_2_1_1_3_1_1"/>
( ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> ) or ( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E44">A580</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_15_1_2_1_1_3_1_1_1"/><span class="lab"><font color="Green" title="E57">A615</font></span>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_15_1_2_1_1_3_1_1_1"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1"><span class="lab"><font color="Green" title="E26">A492</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_3"><span class="lab"><font color="Green" title="E56">A614</font></span></a></span>;<br/><a NAME="E3:1_1_1_15_1_2_1_1_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E4:1_1_1_15_1_2_1_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A616</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E19:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E53">A589</font></span></a>, <a class="txt" href="#E14:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E48">A584</font></span></a>, <a class="txt" href="#E18:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E52">A588</font></span></a>, <a class="txt" href="#E11:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E45">A581</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E57">A615</font></span></a>, <a class="ref" href="goboard7.html#T15" title="GOBOARD7:th.15">GOBOARD7:15</a></span>;<br/><a NAME="E5:1_1_1_15_1_2_1_1_3_1_1_1"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1"><span class="lab"><font color="Green" title="E26">A492</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_3"><span class="lab"><font color="Green" title="E56">A614</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E57">A615</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_15_1_2_1_1_3_1_1_1"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_15"><span class="lab"><font color="Green" title="E34">A570</font></span></a>, <a class="txt" href="#E2:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E37">A573</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E4:1_1_1_15_1_2_1_1_3_1_1_1"><span class="lab"><font color="Green" title="E58">A616</font></span></a>, <a class="ref" href="sppol_2.html#T30" title="SPPOL_2:th.30">SPPOL_2:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_15_1_2_1_1_3_1_1_2"/><span class="lab"><font color="Green" title="E57">A617</font></span>: 
( <font color="Maroon" title="c10">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_15_1_2_1_1_3_1_1_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1"><span class="lab"><font color="Green" title="E26">A492</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_3"><span class="lab"><font color="Green" title="E56">A614</font></span></a></span>;<br/><a NAME="E3:1_1_1_15_1_2_1_1_3_1_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E4:1_1_1_15_1_2_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A618</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E19:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E53">A589</font></span></a>, <a class="txt" href="#E20:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E54">A590</font></span></a>, <a class="txt" href="#E21:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E55">A591</font></span></a>, <a class="txt" href="#E13:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E47">A583</font></span></a>, <a class="txt" href="#E11:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E45">A581</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E57">A617</font></span></a>, <a class="ref" href="goboard7.html#T16" title="GOBOARD7:th.16">GOBOARD7:16</a></span>;<br/><a NAME="E5:1_1_1_15_1_2_1_1_3_1_1_2"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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="c13">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1"><span class="lab"><font color="Green" title="E26">A492</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_3"><span class="lab"><font color="Green" title="E56">A614</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E57">A617</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_15_1_2_1_1_3_1_1_2"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_15"><span class="lab"><font color="Green" title="E34">A570</font></span></a>, <a class="txt" href="#E2:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E37">A573</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E4:1_1_1_15_1_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E58">A618</font></span></a>, <a class="ref" href="sppol_2.html#T30" title="SPPOL_2:th.30">SPPOL_2:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_15_1_2_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E57">A619</font></span>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_15_1_2_1_1_3_1_1_3"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c12">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1"><span class="lab"><font color="Green" title="E26">A492</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_3"><span class="lab"><font color="Green" title="E56">A614</font></span></a></span>;<br/><a NAME="E3:1_1_1_15_1_2_1_1_3_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E4:1_1_1_15_1_2_1_1_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A620</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c12">i2</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E17:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E51">A587</font></span></a>, <a class="txt" href="#E14:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E48">A584</font></span></a>, <a class="txt" href="#E15:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E49">A585</font></span></a>, <a class="txt" href="#E11:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E45">A581</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E57">A619</font></span></a>, <a class="ref" href="goboard7.html#T17" title="GOBOARD7:th.17">GOBOARD7:17</a></span>;<br/><a NAME="E5:1_1_1_15_1_2_1_1_3_1_1_3"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1"><span class="lab"><font color="Green" title="E26">A492</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_3"><span class="lab"><font color="Green" title="E56">A614</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E57">A619</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_15_1_2_1_1_3_1_1_3"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_15"><span class="lab"><font color="Green" title="E34">A570</font></span></a>, <a class="txt" href="#E2:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E37">A573</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E4:1_1_1_15_1_2_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E58">A620</font></span></a>, <a class="ref" href="sppol_2.html#T30" title="SPPOL_2:th.30">SPPOL_2:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_2_1_1_3_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_15_1_2_1_1_3_1_1_4"/><span class="lab"><font color="Green" title="E57">A621</font></span>: 
( <font color="Maroon" title="c10">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">i2</font> &amp; <font color="Maroon" title="c11">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><div class="add"><a NAME="E2:1_1_1_15_1_2_1_1_3_1_1_4"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1"><span class="lab"><font color="Green" title="E26">A492</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_3"><span class="lab"><font color="Green" title="E56">A614</font></span></a></span>;<br/><a NAME="E3:1_1_1_15_1_2_1_1_3_1_1_4"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E4:1_1_1_15_1_2_1_1_3_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A622</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E17:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E51">A587</font></span></a>, <a class="txt" href="#E20:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E54">A590</font></span></a>, <a class="txt" href="#E12:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E46">A582</font></span></a>, <a class="txt" href="#E16:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E50">A586</font></span></a>, <a class="txt" href="#E11:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E45">A581</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E57">A621</font></span></a>, <a class="ref" href="goboard7.html#T18" title="GOBOARD7:th.18">GOBOARD7:18</a></span>;<br/><a NAME="E5:1_1_1_15_1_2_1_1_3_1_1_4"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><font color="Maroon" title="c12">i2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c13">j2</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1"><span class="lab"><font color="Green" title="E26">A492</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E36">A572</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E6:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E40">A576</font></span></a>, <a class="txt" href="#E7:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E41">A577</font></span></a>, <a class="txt" href="#E8:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E42">A578</font></span></a>, <a class="txt" href="#E9:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E43">A579</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_3"><span class="lab"><font color="Green" title="E56">A614</font></span></a>, <a class="txt" href="#E1:1_1_1_15_1_2_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E57">A621</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_15_1_2_1_1_3_1_1_4"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_15"><span class="lab"><font color="Green" title="E34">A570</font></span></a>, <a class="txt" href="#E2:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E37">A573</font></span></a>, <a class="txt" href="#E3:1_1_1_15_1_2"><span class="lab"><font color="Green" title="E38">A574</font></span></a>, <a class="txt" href="#E4:1_1_1_15_1_2_1_1_3_1_1_4"><span class="lab"><font color="Green" title="E58">A622</font></span></a>, <a class="ref" href="sppol_2.html#T30" title="SPPOL_2:th.30">SPPOL_2:30</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_15_1_2_1_1_3"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E23:1_1_1_15_1_2"/>
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_16"><span class="kw">now </span></a><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>8</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> $1 is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> ;<br/><span class="kw">assume </span><a NAME="E1:1_1_1_16"/><span class="lab"><font color="Green" title="E34">A623</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 holds <br/> for <font color="Olive" title="b2">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c7">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:1_1_1_16"/><span class="lab"><font color="Green" title="E35">A624</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>8</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>8</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_16_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c8">k</font>] implies <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_1_16_1"/><span class="lab"><font color="Green" title="E36">A625</font></span>: 
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_16_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">n</font>, <font color="Maroon" title="c10">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon" title="c10">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">m</font> implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font> )</span><br/><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_16_1_1"/><span class="lab"><font color="Green" title="E37">A626</font></span>: 
<font color="Maroon" title="c9">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_16_1_1"/><span class="lab"><font color="Green" title="E38">A627</font></span>: 
<font color="Maroon" title="c10">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_16_1_1"/><span class="lab"><font color="Green" title="E39">A628</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><a NAME="E4:1_1_1_16_1_1"/><span class="lab"><font color="Green" title="E40">A629</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E37">A626</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:1_1_1_16_1_1"/><span class="lab"><font color="Green" title="E41">A630</font></span>: 
<font color="Maroon" title="c10">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E38">A627</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E6:1_1_1_16_1_1"/><span class="lab"><font color="Green" title="E42">A631</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E38">A627</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E7:1_1_1_16_1_1"/><span class="lab"><font color="Green" title="E43">A632</font></span>: 
<font color="Maroon" title="c9">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E37">A626</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E8:1_1_1_16_1_1"/><span class="lab"><font color="Green" title="E44">A633</font></span>: 
 ex <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> &amp; <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_1"><span class="lab"><font color="Green" title="E29">A519</font></span></a></span>;<br/><a NAME="E9:1_1_1_16_1_1"/><span class="lab"><font color="Green" title="E45">A634</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/><a NAME="E10:1_1_1_16_1_1"/><span class="lab"><font color="Green" title="E46">A635</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_16_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_16_1_1_1"/>
( ( <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> ) or ( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> ) or ( <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E43">A632</font></span></a>, <a class="txt" href="#E5:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E41">A630</font></span></a>, <a class="txt" href="#E10:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E46">A635</font></span></a>, <a class="ref" href="nat_1.html#T8" title="NAT_1:th.8">NAT_1:8</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_16_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_16_1_1_1_1"/><span class="lab"><font color="Green" title="E47">A636</font></span>: 
( <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><a NAME="E2:1_1_1_16_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A637</font></span>: 
<font color="Maroon" title="c10">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E42">A631</font></span></a>, <a class="txt" href="#E9:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E45">A634</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E3:1_1_1_16_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A638</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E44">A633</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><a NAME="E4:1_1_1_16_1_1_1_1"/><span class="lab"><font color="Green" title="E50">A639</font></span>: 
<font color="Maroon" title="c9">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E40">A629</font></span></a>, <a class="txt" href="#E9:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E45">A634</font></span></a>, <a class="txt" href="#E1:1_1_1_16_1_1_1_1"><span class="lab"><font color="Green" title="E47">A636</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:1_1_1_16_1_1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c9">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E44">A633</font></span></a>, <a class="ref" href="finseq_4.html#T68" title="FINSEQ_4:th.68">FINSEQ_4:68</a></span>;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_16_1_1_1_1"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_16_1"><span class="lab"><font color="Green" title="E36">A625</font></span></a>, <a class="txt" href="#E3:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E39">A628</font></span></a>, <a class="txt" href="#E4:1_1_1_16_1_1_1_1"><span class="lab"><font color="Green" title="E50">A639</font></span></a>, <a class="txt" href="#E2:1_1_1_16_1_1_1_1"><span class="lab"><font color="Green" title="E48">A637</font></span></a>, <a class="txt" href="#E3:1_1_1_16_1_1_1_1"><span class="lab"><font color="Green" title="E49">A638</font></span></a>, <a class="ref" href="partfun2.html#T10" title="PARTFUN2:th.10">PARTFUN2:10</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_16_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_16_1_1_1_2"/>
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_16_1_1_1_2"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_16"><span class="lab"><font color="Green" title="E34">A623</font></span></a>, <a class="txt" href="#E2:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E38">A627</font></span></a>, <a class="txt" href="#E3:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E39">A628</font></span></a>, <a class="txt" href="#E10:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E46">A635</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_16_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_16_1_1_1_3"/>
( <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">k</font> &amp; <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_16_1_1_1_3"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_16"><span class="lab"><font color="Green" title="E34">A623</font></span></a>, <a class="txt" href="#E1:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E37">A626</font></span></a>, <a class="txt" href="#E3:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E39">A628</font></span></a>, <a class="txt" href="#E10:1_1_1_16_1_1"><span class="lab"><font color="Green" title="E46">A635</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_16_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_16_1_1_1_4"/>
( <font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive">b<sub>2</sub></font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_16_1_1_1_4"/>
<font color="Maroon" title="c9">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:1_1_1_16_1"/>
<font color="Maroon">S<sub>8</sub></font>[<font color="Maroon" title="c8">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="partfun2.html#T9" title="PARTFUN2:th.9">PARTFUN2:9</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E3:1_1_1_16"/><span class="lab"><font color="Green" title="E36">A640</font></span>: 
<font color="Maroon">S<sub>8</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1"><span class="lab"><font color="Green" title="E9">A155</font></span></a></span>;<br/><a NAME="E4:1_1_1_16"/><span class="lab"><font color="Green" title="E37">A641</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>8</sub></font>[<font color="Olive" title="b1">k</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E3:1_1_1_16"><span class="lab"><font color="Green" title="E36">A640</font></span></a>, <a class="txt" href="#E2:1_1_1_16"><span class="lab"><font color="Green" title="E35">A624</font></span></a>);<br/></span><a NAME="E5:1_1_1_16"/><span class="lab"><font color="Green" title="E38">A642</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds   <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">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="1_1_1_16_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">k</font></span><br/>

<a NAME="E1:1_1_1_16_2"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_16"><span class="lab"><font color="Green" title="E37">A641</font></span></a></span>;<br/>
<span class="kw">hence </span> <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T62" title="FINSEQ_4:th.62">FINSEQ_4:62</a></span>
<br/>.= 
<font color="Maroon" title="c8">k</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">reconsider </span><font color="Maroon" title="c8">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E7:1_1_1_16"/><span class="lab"><font color="Green" title="E39">A643</font></span>: 
<font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E8:1_1_1_16"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a></span>;<br/><a NAME="E9:1_1_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A644</font></span>: 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#T8" title="GOBRD13:th.8">GOBRD13:8</a>, <a class="ref" href="nat_1.html#T43" title="NAT_1:th.43">NAT_1:43</a></span>;<br/><a NAME="E10:1_1_1_16"/>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd13.html#T7" title="GOBRD13:th.7">GOBRD13:7</a></span>;<br/><a NAME="E11:1_1_1_16"/><span class="kw">then </span>
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c8">k</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_16"><span class="lab"><font color="Green" title="E40">A644</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">hence </span><a NAME="E12:1_1_1_16"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_16"><span class="lab"><font color="Green" title="E38">A642</font></span></a>, <a class="txt" href="#E7:1_1_1_16"><span class="lab"><font color="Green" title="E39">A643</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E37:1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A645</font></span>: 
 ex <font color="Olive" title="b1">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">k</font>]
 
;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">k</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><span class="kw"> such that </span><br/><a NAME="E39:1_1_1"/><span class="lab"><font color="Green" title="E35">A646</font></span>: 
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon" title="c8">k</font>]
 <span class="kw">and </span><br/><a NAME="E40:1_1_1"/><span class="lab"><font color="Green" title="E36">A647</font></span>: 
 for <font color="Olive" title="b1">l</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive" title="b1">l</font>] holds <br/><font color="Maroon" title="c8">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">l</font>
 <span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S5" title="NAT_1:sch.5">NAT_1:sch 5</a>(<a class="txt" href="#E37:1_1_1"><span class="lab"><font color="Green" title="E34">A645</font></span></a>);<br/></span>
<span class="kw">reconsider </span><font color="Maroon" title="c9">k</font> = <font color="Maroon" title="c8">k</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E43:1_1_1"/><span class="lab"><font color="Green" title="E37">A648</font></span>: 
<font color="Maroon" title="c10">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E44:1_1_1"/><span class="lab"><font color="Green" title="E38">A649</font></span>: 
<font color="Maroon" title="c10">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E45:1_1_1"/><span class="lab"><font color="Green" title="E39">A650</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:1_1_1"><span class="lab"><font color="Green" title="E35">A646</font></span></a></span>;<br/>
<a NAME="E46:1_1_1"/><span class="lab"><font color="Green" title="E40">A651</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:1_1_1"><span class="lab"><font color="Green" title="E37">A648</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">f</font> = <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">k</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:1_1_1"><span class="lab"><font color="Green" title="E35">A646</font></span></a></span>;<br/>
<a NAME="E48:1_1_1"/><span class="lab"><font color="Green" title="E41">A652</font></span>: 
<font color="Maroon" title="c11">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a></span>;<br/>
<a NAME="E49:1_1_1"/><span class="lab"><font color="Green" title="E42">A653</font></span>: 
<font color="Maroon" title="c10">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="c11">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:1_1_1"><span class="lab"><font color="Green" title="E37">A648</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E50:1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A654</font></span>: 
<font color="Maroon" title="c10">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="c11">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E44:1_1_1"><span class="lab"><font color="Green" title="E38">A649</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E51:1_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:1_1_1"><span class="lab"><font color="Green" title="E40">A651</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E52:1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A655</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c11">f</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E53:1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E45">A656</font></span>: 
<font color="Maroon" title="c9">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/>
<a NAME="E54:1_1_1"/><span class="lab"><font color="Green" title="E46">A657</font></span>: 
<font color="Maroon">S<sub>7</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1"><span class="lab"><font color="Green" title="E9">A155</font></span></a>, <a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a>, <a class="ref" href="sppol_2.html#T26" title="SPPOL_2:th.26">SPPOL_2:26</a></span>;<br/>
<a NAME="E55:1_1_1"/>
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>7</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E54:1_1_1"><span class="lab"><font color="Green" title="E46">A657</font></span></a>, <a class="txt" href="#E35:1_1_1"><span class="lab"><font color="Green" title="E33">A569</font></span></a>);<br/></span>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c12">f</font> = <font color="Maroon" title="c11">f</font> as   non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>   <a href="topreal1.html#V2" title="TOPREAL1:attr.2">unfolded</a>  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E52:1_1_1"><span class="lab"><font color="Green" title="E44">A655</font></span></a>, <a class="ref" href="jordan8.html#T4" title="JORDAN8:th.4">JORDAN8:4</a>, <a class="ref" href="jordan8.html#T5" title="JORDAN8:th.5">JORDAN8:5</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">g</font> = <font color="Maroon" title="c12">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>;<br/>
<a NAME="E57:1_1_1"/><span class="lab"><font color="Green" title="E47">A658</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E50:1_1_1"><span class="lab"><font color="Green" title="E43">A654</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E58:1_1_1"/><span class="lab"><font color="Green" title="E48">A659</font></span>: 
 for <font color="Olive" title="b1">h</font> being   non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>  st  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c12">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#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  holds <br/> for <font color="Olive" title="b3">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">n</font> &amp; <font color="Olive" title="b3">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">Comp</font> &amp;  not <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Olive" title="b1">h</font> holds <br/><font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b2">Comp</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_17"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">h</font> be   non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c12">f</font> implies  for <font color="Olive" title="b1">Comp</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">Comp</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  holds <br/> for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Comp</font> &amp;  not <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font> holds <br/><font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Comp</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_1_17"/><span class="lab"><font color="Green" title="E49">A660</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c12">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">Comp</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive" title="b1">Comp</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  holds <br/> for <font color="Olive" title="b2">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">n</font> &amp; <font color="Olive" title="b2">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Comp</font> &amp;  not <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font> holds <br/><font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Olive" title="b1">Comp</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c15">Comp</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">Comp</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Comp</font> &amp;  not <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font> holds <br/><font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c15">Comp</font> )</span><br/>

<span class="kw">assume </span><a NAME="E2:1_1_1_17"/><span class="lab"><font color="Green" title="E50">A661</font></span>: 
<font color="Maroon" title="c15">Comp</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Comp</font> &amp;  not <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font> holds <br/><font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c15">Comp</font></span><br/>

<span class="kw">let </span><font color="Maroon" title="c16">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font> &amp; <font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font> &amp; <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Comp</font> &amp;  not <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font> implies <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c15">Comp</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:1_1_1_17"/><span class="lab"><font color="Green" title="E51">A662</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n</font>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_17"/><span class="lab"><font color="Green" title="E52">A663</font></span>: 
<font color="Maroon" title="c16">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_17"/><span class="lab"><font color="Green" title="E53">A664</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">Comp</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1_1_17"/><span class="lab"><font color="Green" title="E54">A665</font></span>: 
 not <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c15">Comp</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c17">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c16">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font></span>)</span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c16">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">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="E8:1_1_1_17"/><span class="lab"><font color="Green" title="E55">A666</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c16">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c16">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T16" title="TOPS_1:th.16">TOPS_1:16</a></span>;<br/>
<a NAME="E9:1_1_1_17"/>
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</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="c12">f</font>,<font color="Maroon" title="c16">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E3:1_1_1_17"><span class="lab"><font color="Green" title="E51">A662</font></span></a>, <a class="txt" href="#E4:1_1_1_17"><span class="lab"><font color="Green" title="E52">A663</font></span></a>, <a class="ref" href="jordan9.html#T8" title="JORDAN9:th.8">JORDAN9:8</a></span>;<br/>
<a NAME="E10:1_1_1_17"/><span class="kw">then </span>
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">rc</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_17"><span class="lab"><font color="Green" title="E54">A665</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E11:1_1_1_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A667</font></span>: 
<font color="Maroon" title="c18">rc</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c15">Comp</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_17"><span class="lab"><font color="Green" title="E53">A664</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<a NAME="E12:1_1_1_17"/><span class="lab"><font color="Green" title="E57">A668</font></span>: 
<font color="Maroon" title="c18">rc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c16">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T13" title="SUBSET_1:th.13">SUBSET_1:13</a></span>;<br/>
<a NAME="E13:1_1_1_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A669</font></span>: 
<font color="Maroon" title="c18">rc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></span>;<br/>
<a NAME="E14:1_1_1_17"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c16">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c12">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E3:1_1_1_17"><span class="lab"><font color="Green" title="E51">A662</font></span></a>, <a class="txt" href="#E4:1_1_1_17"><span class="lab"><font color="Green" title="E52">A663</font></span></a>, <a class="ref" href="jordan9.html#T15" title="JORDAN9:th.15">JORDAN9:15</a></span>;<br/>
<a NAME="E15:1_1_1_17"/><span class="kw">then </span>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c16">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_17"><span class="lab"><font color="Green" title="E49">A660</font></span></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/>
<a NAME="E16:1_1_1_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A670</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c16">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T23" title="SUBSET_1:th.23">SUBSET_1:23</a></span>;<br/>
<a NAME="E17:1_1_1_17"/>
<font color="Maroon" title="c18">rc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c16">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<a NAME="E18:1_1_1_17"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A671</font></span>: 
<font color="Maroon" title="c18">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c16">n</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E3:1_1_1_17"><span class="lab"><font color="Green" title="E51">A662</font></span></a>, <a class="txt" href="#E4:1_1_1_17"><span class="lab"><font color="Green" title="E52">A663</font></span></a>, <a class="ref" href="jordan9.html#T11" title="JORDAN9:th.11">JORDAN9:11</a></span>;<br/>
<a NAME="E19:1_1_1_17"/><span class="lab"><font color="Green" title="E61">A672</font></span>: 
<font color="Maroon" title="c18">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="1_1_1_17_1"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:1_1_1_17_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c16">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E3:1_1_1_17"><span class="lab"><font color="Green" title="E51">A662</font></span></a>, <a class="txt" href="#E4:1_1_1_17"><span class="lab"><font color="Green" title="E52">A663</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:1_1_1_17_1"/><span class="lab"><font color="Green" title="E62">A673</font></span>: 
<font color="Maroon" title="c19">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="c12">f</font>,<font color="Maroon" title="c16">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_17_1"/><span class="lab"><font color="Green" title="E63">A674</font></span>: 
<font color="Maroon" title="c19">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c20">p</font> = <font color="Maroon" title="c19">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_17_1"><span class="lab"><font color="Green" title="E62">A673</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_17_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">take </span><font color="Maroon" title="c21">p</font> = <font color="Maroon" title="c20">p</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">rc</font> &amp; <font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> )</span><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_17_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:1_1_1_17_1_1_1"/>
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c14">h</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:1_1_1_17_1_1_1"/><span class="lab"><font color="Green" title="E64">A675</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">j</font>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_17_1_1_1"/><span class="lab"><font color="Green" title="E65">A676</font></span>: 
<font color="Maroon" title="c22">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_17_1_1_1"/><span class="lab"><font color="Green" title="E66">A677</font></span>: 
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c22">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_17"><span class="lab"><font color="Green" title="E49">A660</font></span></a>, <a class="ref" href="sppol_2.html#T13" title="SPPOL_2:th.13">SPPOL_2:13</a></span>;<br/><a NAME="E6:1_1_1_17_1_1_1"/>
<font color="Maroon" title="c21">p</font> <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> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c22">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c22">j</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E3:1_1_1_17_1_1_1"><span class="lab"><font color="Green" title="E64">A675</font></span></a>, <a class="txt" href="#E4:1_1_1_17_1_1_1"><span class="lab"><font color="Green" title="E65">A676</font></span></a>, <a class="txt" href="#E5:1_1_1_17_1_1_1"><span class="lab"><font color="Green" title="E66">A677</font></span></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>;<br/><a NAME="E7:1_1_1_17_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A678</font></span>: 
<font color="Maroon" title="c21">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="c12">f</font>,<font color="Maroon" title="c22">j</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E8:1_1_1_17_1_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c22">j</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E3:1_1_1_17_1_1_1"><span class="lab"><font color="Green" title="E64">A675</font></span></a>, <a class="txt" href="#E4:1_1_1_17_1_1_1"><span class="lab"><font color="Green" title="E65">A676</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E9:1_1_1_17_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_17_1"><span class="lab"><font color="Green" title="E63">A674</font></span></a>, <a class="txt" href="#E7:1_1_1_17_1_1_1"><span class="lab"><font color="Green" title="E67">A678</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E2:1_1_1_17_1_1"/>
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c18">rc</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_17_1"><span class="lab"><font color="Green" title="E62">A673</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font></span><br/><span class="kw">thus </span><a NAME="E3:1_1_1_17_1_1"/>
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_17_1"><span class="lab"><font color="Green" title="E63">A674</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:1_1_1_17_1"/>
<font color="Maroon" title="c18">rc</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E20:1_1_1_17"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c16">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E3:1_1_1_17"><span class="lab"><font color="Green" title="E51">A662</font></span></a>, <a class="txt" href="#E4:1_1_1_17"><span class="lab"><font color="Green" title="E52">A663</font></span></a>, <a class="ref" href="jordan9.html#T10" title="JORDAN9:th.10">JORDAN9:10</a></span>;<br/>
<a NAME="E21:1_1_1_17"/><span class="kw">then </span>
<font color="Maroon" title="c18">rc</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_1_17"><span class="lab"><font color="Green" title="E57">A668</font></span></a>, <a class="txt" href="#E16:1_1_1_17"><span class="lab"><font color="Green" title="E59">A670</font></span></a>, <a class="txt" href="#E8:1_1_1_17"><span class="lab"><font color="Green" title="E55">A666</font></span></a>, <a class="txt" href="#E18:1_1_1_17"><span class="lab"><font color="Green" title="E60">A671</font></span></a>, <a class="ref" href="connsp_1.html#T18" title="CONNSP_1:th.18">CONNSP_1:18</a>, <a class="ref" href="xboole_1.html#T19" title="XBOOLE_1:th.19">XBOOLE_1:19</a></span>;<br/>
<a NAME="E22:1_1_1_17"/><span class="kw">then </span>
<font color="Maroon" title="c18">rc</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c15">Comp</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_17"><span class="lab"><font color="Green" title="E50">A661</font></span></a>, <a class="txt" href="#E11:1_1_1_17"><span class="lab"><font color="Green" title="E56">A667</font></span></a>, <a class="txt" href="#E13:1_1_1_17"><span class="lab"><font color="Green" title="E58">A669</font></span></a>, <a class="ref" href="goboard9.html#T4" title="GOBOARD9:th.4">GOBOARD9:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E23:1_1_1_17"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c15">Comp</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1_17"><span class="lab"><font color="Green" title="E61">A672</font></span></a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E59:1_1_1"/><span class="lab"><font color="Green" title="E49">A679</font></span>: 
 for <font color="Olive" title="b1">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font> holds <br/> <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">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="1_1_1_18"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c14">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font> &amp; <font color="Maroon" title="c14">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font> implies  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span>)</span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_18"/><span class="lab"><font color="Green" title="E50">A680</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">i</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_18"/><span class="lab"><font color="Green" title="E51">A681</font></span>: 
<font color="Maroon" title="c14">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span>)</span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c15">i1</font>, <font color="Maroon" title="c16">j1</font>, <font color="Maroon" title="c17">i2</font>, <font color="Maroon" title="c18">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E4:1_1_1_18"/><span class="lab"><font color="Green" title="E52">A682</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c15">i1</font>,<font color="Maroon" title="c16">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_18"/><span class="lab"><font color="Green" title="E53">A683</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c14">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="c15">i1</font>,<font color="Maroon" title="c16">j1</font>)
 <span class="kw">and </span><br/><a NAME="E6:1_1_1_18"/><span class="lab"><font color="Green" title="E54">A684</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">i2</font>,<font color="Maroon" title="c18">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E7:1_1_1_18"/><span class="lab"><font color="Green" title="E55">A685</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c17">i2</font>,<font color="Maroon" title="c18">j2</font>)
 <span class="kw">and </span><br/><a NAME="E8:1_1_1_18"/><span class="lab"><font color="Green" title="E56">A686</font></span>: 
( ( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> ) or ( <font color="Maroon" title="c15">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> ) or ( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> ) or ( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E1:1_1_1_18"><span class="lab"><font color="Green" title="E50">A680</font></span></a>, <a class="txt" href="#E2:1_1_1_18"><span class="lab"><font color="Green" title="E51">A681</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/>
<a NAME="E9:1_1_1_18"/><span class="lab"><font color="Green" title="E57">A687</font></span>: 
<font color="Maroon" title="c15">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_18"><span class="lab"><font color="Green" title="E52">A682</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E10:1_1_1_18"/><span class="lab"><font color="Green" title="E58">A688</font></span>: 
<font color="Maroon" title="c17">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_18"><span class="lab"><font color="Green" title="E54">A684</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E11:1_1_1_18"/><span class="lab"><font color="Green" title="E59">A689</font></span>: 
<font color="Maroon" title="c15">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c15">i1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E12:1_1_1_18"/><span class="lab"><font color="Green" title="E60">A690</font></span>: 
<font color="Maroon" title="c16">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_18"><span class="lab"><font color="Green" title="E52">A682</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E13:1_1_1_18"/><span class="lab"><font color="Green" title="E61">A691</font></span>: 
<font color="Maroon" title="c16">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c16">j1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E14:1_1_1_18"/><span class="lab"><font color="Green" title="E62">A692</font></span>: 
<font color="Maroon" title="c18">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_18"><span class="lab"><font color="Green" title="E54">A684</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E15:1_1_1_18"/><span class="lab"><font color="Green" title="E63">A693</font></span>: 
<font color="Maroon" title="c17">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c17">i2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E16:1_1_1_18"/><span class="lab"><font color="Green" title="E64">A694</font></span>: 
<font color="Maroon" title="c18">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c18">j2</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_18_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_18_1"/>
( ( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> ) or ( <font color="Maroon" title="c15">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> ) or ( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> ) or ( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_18"><span class="lab"><font color="Green" title="E56">A686</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_18_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_18_1_1"/><span class="lab"><font color="Green" title="E65">A695</font></span>: 
( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_18_1_1"/><span class="lab"><font color="Green" title="E66">A696</font></span>: 
<font color="Maroon" title="c15">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_18"><span class="lab"><font color="Green" title="E57">A687</font></span></a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/><a NAME="E3:1_1_1_18_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">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="c15">i1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c16">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E1:1_1_1_18"><span class="lab"><font color="Green" title="E50">A680</font></span></a>, <a class="txt" href="#E2:1_1_1_18"><span class="lab"><font color="Green" title="E51">A681</font></span></a>, <a class="txt" href="#E4:1_1_1_18"><span class="lab"><font color="Green" title="E52">A682</font></span></a>, <a class="txt" href="#E5:1_1_1_18"><span class="lab"><font color="Green" title="E53">A683</font></span></a>, <a class="txt" href="#E6:1_1_1_18"><span class="lab"><font color="Green" title="E54">A684</font></span></a>, <a class="txt" href="#E7:1_1_1_18"><span class="lab"><font color="Green" title="E55">A685</font></span></a>, <a class="txt" href="#E13:1_1_1_18"><span class="lab"><font color="Green" title="E61">A691</font></span></a>, <a class="txt" href="#E16:1_1_1_18"><span class="lab"><font color="Green" title="E64">A694</font></span></a>, <a class="txt" href="#E1:1_1_1_18_1_1"><span class="lab"><font color="Green" title="E65">A695</font></span></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_18_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:1_1_1_18"><span class="lab"><font color="Green" title="E60">A690</font></span></a>, <a class="txt" href="#E2:1_1_1_18_1_1"><span class="lab"><font color="Green" title="E66">A696</font></span></a>, <a class="ref" href="gobrd11.html#T35" title="GOBRD11:th.35">GOBRD11:35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_18_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_18_1_2"/>
( <font color="Maroon" title="c15">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_18_1_2"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">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="c15">i1</font>,<font color="Maroon" title="c16">j1</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E1:1_1_1_18"><span class="lab"><font color="Green" title="E50">A680</font></span></a>, <a class="txt" href="#E2:1_1_1_18"><span class="lab"><font color="Green" title="E51">A681</font></span></a>, <a class="txt" href="#E4:1_1_1_18"><span class="lab"><font color="Green" title="E52">A682</font></span></a>, <a class="txt" href="#E5:1_1_1_18"><span class="lab"><font color="Green" title="E53">A683</font></span></a>, <a class="txt" href="#E6:1_1_1_18"><span class="lab"><font color="Green" title="E54">A684</font></span></a>, <a class="txt" href="#E7:1_1_1_18"><span class="lab"><font color="Green" title="E55">A685</font></span></a>, <a class="txt" href="#E11:1_1_1_18"><span class="lab"><font color="Green" title="E59">A689</font></span></a>, <a class="txt" href="#E15:1_1_1_18"><span class="lab"><font color="Green" title="E63">A693</font></span></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:1_1_1_18_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_18"><span class="lab"><font color="Green" title="E57">A687</font></span></a>, <a class="txt" href="#E12:1_1_1_18"><span class="lab"><font color="Green" title="E60">A690</font></span></a>, <a class="ref" href="gobrd11.html#T35" title="GOBRD11:th.35">GOBRD11:35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_18_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_18_1_3"/><span class="lab"><font color="Green" title="E65">A697</font></span>: 
( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_18_1_3"/><span class="lab"><font color="Green" title="E66">A698</font></span>: 
<font color="Maroon" title="c18">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_18"><span class="lab"><font color="Green" title="E62">A692</font></span></a>, <a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/><a NAME="E3:1_1_1_18_1_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">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">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c18">j2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E1:1_1_1_18"><span class="lab"><font color="Green" title="E50">A680</font></span></a>, <a class="txt" href="#E2:1_1_1_18"><span class="lab"><font color="Green" title="E51">A681</font></span></a>, <a class="txt" href="#E4:1_1_1_18"><span class="lab"><font color="Green" title="E52">A682</font></span></a>, <a class="txt" href="#E5:1_1_1_18"><span class="lab"><font color="Green" title="E53">A683</font></span></a>, <a class="txt" href="#E6:1_1_1_18"><span class="lab"><font color="Green" title="E54">A684</font></span></a>, <a class="txt" href="#E7:1_1_1_18"><span class="lab"><font color="Green" title="E55">A685</font></span></a>, <a class="txt" href="#E11:1_1_1_18"><span class="lab"><font color="Green" title="E59">A689</font></span></a>, <a class="txt" href="#E15:1_1_1_18"><span class="lab"><font color="Green" title="E63">A693</font></span></a>, <a class="txt" href="#E1:1_1_1_18_1_3"><span class="lab"><font color="Green" title="E65">A697</font></span></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_18_1_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1_18"><span class="lab"><font color="Green" title="E58">A688</font></span></a>, <a class="txt" href="#E2:1_1_1_18_1_3"><span class="lab"><font color="Green" title="E66">A698</font></span></a>, <a class="ref" href="gobrd11.html#T35" title="GOBRD11:th.35">GOBRD11:35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_18_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_18_1_4"/>
( <font color="Maroon" title="c15">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i2</font> &amp; <font color="Maroon" title="c16">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_18_1_4"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">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="c15">i1</font>,<font color="Maroon" title="c18">j2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E1:1_1_1_18"><span class="lab"><font color="Green" title="E50">A680</font></span></a>, <a class="txt" href="#E2:1_1_1_18"><span class="lab"><font color="Green" title="E51">A681</font></span></a>, <a class="txt" href="#E4:1_1_1_18"><span class="lab"><font color="Green" title="E52">A682</font></span></a>, <a class="txt" href="#E5:1_1_1_18"><span class="lab"><font color="Green" title="E53">A683</font></span></a>, <a class="txt" href="#E6:1_1_1_18"><span class="lab"><font color="Green" title="E54">A684</font></span></a>, <a class="txt" href="#E7:1_1_1_18"><span class="lab"><font color="Green" title="E55">A685</font></span></a>, <a class="txt" href="#E13:1_1_1_18"><span class="lab"><font color="Green" title="E61">A691</font></span></a>, <a class="txt" href="#E16:1_1_1_18"><span class="lab"><font color="Green" title="E64">A694</font></span></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:1_1_1_18_1_4"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c14">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_18"><span class="lab"><font color="Green" title="E57">A687</font></span></a>, <a class="txt" href="#E14:1_1_1_18"><span class="lab"><font color="Green" title="E62">A692</font></span></a>, <a class="ref" href="gobrd11.html#T35" title="GOBRD11:th.35">GOBRD11:35</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E60:1_1_1"/>
<font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/>
<a NAME="E61:1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E62:1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A699</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:1_1_1"><span class="lab"><font color="Green" title="E47">A658</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E63:1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A700</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="rfinseq.html#D1" title="RFINSEQ:def.1">RFINSEQ:def 1</a></span>;<br/>
<a NAME="E64:1_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:1_1_1"><span class="lab"><font color="Green" title="E50">A699</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">g</font> = <font color="Maroon" title="c12">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T27" title="CARD_1:th.27">CARD_1:27</a></span>;<br/>
<a NAME="E66:1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">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="c14">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E52">A701</font></span>: <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span> = 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T27" title="FINSEQ_5:th.27">FINSEQ_5:27</a></span>
<br/>.= 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:1_1_1"><span class="lab"><font color="Green" title="E51">A700</font></span></a></span>
;<br/>
<a NAME="E68:1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E57:1_1_1"><span class="lab"><font color="Green" title="E47">A658</font></span></a>, <a class="txt" href="#E63:1_1_1"><span class="lab"><font color="Green" title="E51">A700</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E69:1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A702</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:1_1_1"><span class="lab"><font color="Green" title="E40">A651</font></span></a>, <a class="ref" href="xreal_1.html#T233" title="XREAL_1:th.233">XREAL_1:233</a></span>;<br/>
<a NAME="E70:1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A703</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font>
 
;<br/>
<a NAME="E71:1_1_1"/><span class="lab"><font color="Green" title="E55">A704</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="ref" href="jordan8.html#T2" title="JORDAN8:th.2">JORDAN8:2</a></span>;<br/>
<a NAME="E72:1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A705</font></span>: 
<font color="Maroon" title="c14">g</font> is  <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan8.html#T4" title="JORDAN8:th.4">JORDAN8:4</a></span>;<br/>
<a NAME="E73:1_1_1"/><span class="lab"><font color="Green" title="E57">A706</font></span>: 
 not <font color="Maroon" title="c14">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="1_1_1_20"><span class="kw">proof </span></a><div class="add">

<span class="kw">take </span>
1
; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="seqm_3.html#D10" title="SEQM_3:def.10">SEQM_3:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g</font> &amp;  not <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> )</span><br/>

<span class="kw">take </span>
2
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g</font> &amp; 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g</font> &amp;  not <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 )</span><br/>

<span class="kw">thus </span><a NAME="E1:1_1_1_20"/><span class="lab"><font color="Green" title="E58">A707</font></span>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g</font> &amp;  not <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2 )</span><br/>

<span class="kw">thus </span><a NAME="E2:1_1_1_20"/><span class="lab"><font color="Green" title="E59">A708</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c14">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:1_1_1"><span class="lab"><font color="Green" title="E53">A702</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2</span><br/>

<a NAME="E3:1_1_1_20"/><span class="kw">then </span>
<font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E72:1_1_1"><span class="lab"><font color="Green" title="E56">A705</font></span></a>, <a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a>, <a class="ref" href="goboard7.html#T29" title="GOBOARD7:th.29">GOBOARD7:29</a></span>;<br/>
<a NAME="E4:1_1_1_20"/><span class="kw">then </span>
<font color="Maroon" title="c14">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="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_20"><span class="lab"><font color="Green" title="E58">A707</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:1_1_1_20"/>
 not <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 2
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_20"><span class="lab"><font color="Green" title="E59">A708</font></span></a>, <a class="ref" href="partfun1.html#D6" title="PARTFUN1:def.6">PARTFUN1:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E74:1_1_1"/><span class="lab"><font color="Green" title="E58">A709</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/>
<a NAME="E75:1_1_1"/><span class="lab"><font color="Green" title="E59">A710</font></span>: 
 for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">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="c14">g</font> &amp; <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font> holds <br/><font color="Olive" title="b2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_21"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">j</font>, <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <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="c14">g</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Maroon" title="c15">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="c14">g</font> &amp; <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j</font> implies <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">j</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_21"/><span class="lab"><font color="Green" title="E60">A711</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_21"/><span class="lab"><font color="Green" title="E61">A712</font></span>: 
<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="c14">g</font>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_21"/><span class="lab"><font color="Green" title="E62">A713</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">j</font>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_21"/><span class="lab"><font color="Green" title="E63">A714</font></span>: 
<font color="Maroon" title="c15">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="c14">g</font>
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_21"/><span class="lab"><font color="Green" title="E64">A715</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1_1_21"/><span class="lab"><font color="Green" title="E65">A716</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E7:1_1_1_21"/><span class="lab"><font color="Green" title="E66">A717</font></span>: 
<font color="Maroon" title="c16">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="c14">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_21"><span class="lab"><font color="Green" title="E60">A711</font></span></a>, <a class="txt" href="#E2:1_1_1_21"><span class="lab"><font color="Green" title="E61">A712</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E8:1_1_1_21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A718</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T27" title="FINSEQ_5:th.27">FINSEQ_5:27</a></span>;<br/>
<a NAME="E9:1_1_1_21"/><span class="lab"><font color="Green" title="E68">A719</font></span>: 
<font color="Maroon" title="c15">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="c14">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_21"><span class="lab"><font color="Green" title="E62">A713</font></span></a>, <a class="txt" href="#E4:1_1_1_21"><span class="lab"><font color="Green" title="E63">A714</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E10:1_1_1_21"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A720</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T27" title="FINSEQ_5:th.27">FINSEQ_5:27</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_21_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_21_1"/>
( <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font> or <font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">i</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_21"><span class="lab"><font color="Green" title="E65">A716</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_21_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_21_1_1"/><span class="lab"><font color="Green" title="E70">A721</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c17">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font>;<br/><span class="kw">set </span><font color="Maroon" title="c18">m9</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font>;<br/><a NAME="E2:1_1_1_21_1_1"/><span class="lab"><font color="Green" title="E71">A722</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_21_1_1"><span class="lab"><font color="Green" title="E70">A721</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E3:1_1_1_21_1_1"/><span class="lab"><font color="Green" title="E72">A723</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/><a NAME="E4:1_1_1_21_1_1"/><span class="lab"><font color="Green" title="E73">A724</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E63:1_1_1"><span class="lab"><font color="Green" title="E51">A700</font></span></a>, <a class="txt" href="#E4:1_1_1_21"><span class="lab"><font color="Green" title="E63">A714</font></span></a>, <a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/><a NAME="E5:1_1_1_21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A725</font></span>: 
<font color="Maroon" title="c12">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_1"><span class="lab"><font color="Green" title="E31">A564</font></span></a></span>;<br/><a NAME="E6:1_1_1_21_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E7:1_1_1_21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A726</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_21"><span class="lab"><font color="Green" title="E62">A713</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:1_1_1_21_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">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="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_21_1_1"><span class="lab"><font color="Green" title="E72">A723</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E9:1_1_1_21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A727</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_21_1_1"><span class="lab"><font color="Green" title="E74">A725</font></span></a>, <a class="ref" href="finseq_4.html#T70" title="FINSEQ_4:th.70">FINSEQ_4:70</a></span>;<br/><a NAME="E10:1_1_1_21_1_1"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E11:1_1_1_21_1_1"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_21"><span class="lab"><font color="Green" title="E60">A711</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:1_1_1_21_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A728</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">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="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_21_1_1"><span class="lab"><font color="Green" title="E71">A722</font></span></a>, <a class="txt" href="#E3:1_1_1_21_1_1"><span class="lab"><font color="Green" title="E72">A723</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E13:1_1_1_21_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_21_1_1"><span class="lab"><font color="Green" title="E74">A725</font></span></a>, <a class="ref" href="finseq_4.html#T70" title="FINSEQ_4:th.70">FINSEQ_4:70</a></span>;<br/><span class="kw">hence </span><a NAME="E14:1_1_1_21_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_1"><span class="lab"><font color="Green" title="E36">A647</font></span></a>, <a class="txt" href="#E5:1_1_1_21"><span class="lab"><font color="Green" title="E64">A715</font></span></a>, <a class="txt" href="#E8:1_1_1_21"><span class="lab"><font color="Green" title="E67">A718</font></span></a>, <a class="txt" href="#E9:1_1_1_21"><span class="lab"><font color="Green" title="E68">A719</font></span></a>, <a class="txt" href="#E2:1_1_1_21_1_1"><span class="lab"><font color="Green" title="E71">A722</font></span></a>, <a class="txt" href="#E4:1_1_1_21_1_1"><span class="lab"><font color="Green" title="E73">A724</font></span></a>, <a class="txt" href="#E7:1_1_1_21_1_1"><span class="lab"><font color="Green" title="E75">A726</font></span></a>, <a class="txt" href="#E3:1_1_1_21_1_1"><span class="lab"><font color="Green" title="E72">A723</font></span></a>, <a class="txt" href="#E12:1_1_1_21_1_1"><span class="lab"><font color="Green" title="E77">A728</font></span></a>, <a class="txt" href="#E9:1_1_1_21_1_1"><span class="lab"><font color="Green" title="E76">A727</font></span></a>, <a class="ref" href="finseq_5.html#T27" title="FINSEQ_5:th.27">FINSEQ_5:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_21_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_21_1_2"/><span class="lab"><font color="Green" title="E70">A729</font></span>: 
<font color="Maroon" title="c15">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">i</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c17">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font>;<br/><span class="kw">set </span><font color="Maroon" title="c18">m9</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font>;<br/><a NAME="E2:1_1_1_21_1_2"/><span class="lab"><font color="Green" title="E71">A730</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_21_1_2"><span class="lab"><font color="Green" title="E70">A729</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E3:1_1_1_21_1_2"/><span class="lab"><font color="Green" title="E72">A731</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/><a NAME="E4:1_1_1_21_1_2"/><span class="lab"><font color="Green" title="E73">A732</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E63:1_1_1"><span class="lab"><font color="Green" title="E51">A700</font></span></a>, <a class="txt" href="#E2:1_1_1_21"><span class="lab"><font color="Green" title="E61">A712</font></span></a>, <a class="ref" href="xreal_1.html#T20" title="XREAL_1:th.20">XREAL_1:20</a></span>;<br/><a NAME="E5:1_1_1_21_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A733</font></span>: 
<font color="Maroon" title="c12">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_1"><span class="lab"><font color="Green" title="E31">A564</font></span></a></span>;<br/><a NAME="E6:1_1_1_21_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E7:1_1_1_21_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A734</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_21"><span class="lab"><font color="Green" title="E60">A711</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E8:1_1_1_21_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">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="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_21_1_2"><span class="lab"><font color="Green" title="E72">A731</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E9:1_1_1_21_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A735</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_21_1_2"><span class="lab"><font color="Green" title="E74">A733</font></span></a>, <a class="ref" href="finseq_4.html#T70" title="FINSEQ_4:th.70">FINSEQ_4:70</a></span>;<br/><a NAME="E10:1_1_1_21_1_2"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E11:1_1_1_21_1_2"/><span class="kw">then </span>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_21"><span class="lab"><font color="Green" title="E62">A713</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:1_1_1_21_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A736</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">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="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_21_1_2"><span class="lab"><font color="Green" title="E71">A730</font></span></a>, <a class="txt" href="#E3:1_1_1_21_1_2"><span class="lab"><font color="Green" title="E72">A731</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E13:1_1_1_21_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c16">i</font></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c15">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_21_1_2"><span class="lab"><font color="Green" title="E74">A733</font></span></a>, <a class="ref" href="finseq_4.html#T70" title="FINSEQ_4:th.70">FINSEQ_4:70</a></span>;<br/><span class="kw">hence </span><a NAME="E14:1_1_1_21_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_1"><span class="lab"><font color="Green" title="E36">A647</font></span></a>, <a class="txt" href="#E5:1_1_1_21"><span class="lab"><font color="Green" title="E64">A715</font></span></a>, <a class="txt" href="#E7:1_1_1_21"><span class="lab"><font color="Green" title="E66">A717</font></span></a>, <a class="txt" href="#E10:1_1_1_21"><span class="lab"><font color="Green" title="E69">A720</font></span></a>, <a class="txt" href="#E2:1_1_1_21_1_2"><span class="lab"><font color="Green" title="E71">A730</font></span></a>, <a class="txt" href="#E4:1_1_1_21_1_2"><span class="lab"><font color="Green" title="E73">A732</font></span></a>, <a class="txt" href="#E7:1_1_1_21_1_2"><span class="lab"><font color="Green" title="E75">A734</font></span></a>, <a class="txt" href="#E3:1_1_1_21_1_2"><span class="lab"><font color="Green" title="E72">A731</font></span></a>, <a class="txt" href="#E12:1_1_1_21_1_2"><span class="lab"><font color="Green" title="E77">A736</font></span></a>, <a class="txt" href="#E9:1_1_1_21_1_2"><span class="lab"><font color="Green" title="E76">A735</font></span></a>, <a class="ref" href="finseq_5.html#T27" title="FINSEQ_5:th.27">FINSEQ_5:27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E76:1_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="c14">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E60">A737</font></span>: <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T27" title="FINSEQ_5:th.27">FINSEQ_5:27</a></span>
<br/>.= 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:1_1_1"><span class="lab"><font color="Green" title="E40">A651</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>
;<br/>
<a NAME="E78:1_1_1"/><span class="lab"><font color="Green" title="E61">A738</font></span>: 
 for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font> holds <br/><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_23"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">j</font>, <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Maroon" title="c15">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="c14">g</font> implies <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_23"/><span class="lab"><font color="Green" title="E62">A739</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_23"/><span class="lab"><font color="Green" title="E63">A740</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_23"/><span class="lab"><font color="Green" title="E64">A741</font></span>: 
<font color="Maroon" title="c15">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="c14">g</font>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_23"/><span class="lab"><font color="Green" title="E65">A742</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:1_1_1_23"/><span class="lab"><font color="Green" title="E66">A743</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_23"><span class="lab"><font color="Green" title="E62">A739</font></span></a>, <a class="txt" href="#E2:1_1_1_23"><span class="lab"><font color="Green" title="E63">A740</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E6:1_1_1_23"/><span class="lab"><font color="Green" title="E67">A744</font></span>: 
<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="c14">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_23"><span class="lab"><font color="Green" title="E63">A740</font></span></a>, <a class="txt" href="#E3:1_1_1_23"><span class="lab"><font color="Green" title="E64">A741</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:1_1_1_23"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A745</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_23"><span class="lab"><font color="Green" title="E62">A739</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_23_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_23_1"/>
( <font color="Maroon" title="c15">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="c14">g</font> or <font color="Maroon" title="c15">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="c14">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_23_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_23_1_1"/>
<font color="Maroon" title="c15">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="c14">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_23_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">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="c14">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_23"><span class="lab"><font color="Green" title="E64">A741</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><span class="kw">hence </span><a NAME="E3:1_1_1_23_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_23"><span class="lab"><font color="Green" title="E62">A739</font></span></a>, <a class="txt" href="#E2:1_1_1_23"><span class="lab"><font color="Green" title="E63">A740</font></span></a>, <a class="txt" href="#E4:1_1_1_23"><span class="lab"><font color="Green" title="E65">A742</font></span></a>, <a class="txt" href="#E5:1_1_1_23"><span class="lab"><font color="Green" title="E66">A743</font></span></a>, <a class="txt" href="#E6:1_1_1_23"><span class="lab"><font color="Green" title="E67">A744</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_23_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_23_1_2"/>
<font color="Maroon" title="c15">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="c14">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_23_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:1_1_1"><span class="lab"><font color="Green" title="E39">A650</font></span></a>, <a class="txt" href="#E77:1_1_1"><span class="lab"><font color="Green" title="E60">A737</font></span></a>, <a class="txt" href="#E67:1_1_1"><span class="lab"><font color="Green" title="E52">A701</font></span></a>, <a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_23"><span class="lab"><font color="Green" title="E62">A739</font></span></a>, <a class="txt" href="#E2:1_1_1_23"><span class="lab"><font color="Green" title="E63">A740</font></span></a>, <a class="txt" href="#E4:1_1_1_23"><span class="lab"><font color="Green" title="E65">A742</font></span></a>, <a class="txt" href="#E7:1_1_1_23"><span class="lab"><font color="Green" title="E68">A745</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E79:1_1_1"/><span class="lab"><font color="Green" title="E62">A746</font></span>: 
 for <font color="Olive" title="b1">j</font>, <font color="Olive" title="b2">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">i</font> &amp; <font color="Olive" title="b2">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font> holds <br/><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_24"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">j</font>, <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font> &amp; <font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font> &amp; <font color="Maroon" title="c15">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="c14">g</font> implies <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j</font> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_24"/><span class="lab"><font color="Green" title="E63">A747</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">i</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_24"/><span class="lab"><font color="Green" title="E64">A748</font></span>: 
<font color="Maroon" title="c16">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_24"/><span class="lab"><font color="Green" title="E65">A749</font></span>: 
<font color="Maroon" title="c15">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="c14">g</font>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_24"/><span class="lab"><font color="Green" title="E66">A750</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E5:1_1_1_24"/><span class="lab"><font color="Green" title="E67">A751</font></span>: 
<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="c14">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_24"><span class="lab"><font color="Green" title="E64">A748</font></span></a>, <a class="txt" href="#E3:1_1_1_24"><span class="lab"><font color="Green" title="E65">A749</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E6:1_1_1_24"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_24"><span class="lab"><font color="Green" title="E63">A747</font></span></a>, <a class="txt" href="#E2:1_1_1_24"><span class="lab"><font color="Green" title="E64">A748</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:1_1_1_24"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_24"><span class="lab"><font color="Green" title="E63">A747</font></span></a>, <a class="txt" href="#E2:1_1_1_24"><span class="lab"><font color="Green" title="E64">A748</font></span></a>, <a class="txt" href="#E3:1_1_1_24"><span class="lab"><font color="Green" title="E65">A749</font></span></a>, <a class="txt" href="#E4:1_1_1_24"><span class="lab"><font color="Green" title="E66">A750</font></span></a>, <a class="txt" href="#E5:1_1_1_24"><span class="lab"><font color="Green" title="E67">A751</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E80:1_1_1"/>
<font color="Maroon" title="c14">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="1_1_1_25"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c15">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="goboard5.html#D4" title="GOBOARD5:def.4">GOBOARD5:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive">b<sub>1</sub></font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or ( ( <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">i</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Olive">b<sub>1</sub></font>) )</span><br/><span class="kw">let </span><font color="Maroon" title="c16">j</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c16">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or ( ( <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j</font> ) &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">i</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c16">j</font>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_25"/><span class="lab"><font color="Green" title="E63">A752</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_25"/><span class="lab"><font color="Green" title="E64">A753</font></span>: 
( ( <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &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="c14">g</font> ) or <font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">i</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c16">j</font>)</span><br/>

<a NAME="E3:1_1_1_25"/><span class="lab"><font color="Green" title="E65">A754</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E4:1_1_1_25"/><span class="lab"><font color="Green" title="E66">A755</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E5:1_1_1_25"/><span class="lab"><font color="Green" title="E67">A756</font></span>: 
<font color="Maroon" title="c16">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E6:1_1_1_25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A757</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E7:1_1_1_25"/>
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E8:1_1_1_25"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A758</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1"/>
( ( <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &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="c14">g</font> ) or ( <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font> ) or ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> &amp; <font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_25"><span class="lab"><font color="Green" title="E64">A753</font></span></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E70">A759</font></span>: 
( <font color="Maroon" title="c15">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 &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="c14">g</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">i</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c16">j</font>)</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A760</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E3:1_1_1_25_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A761</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/><a NAME="E4:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E73">A762</font></span>: 
<font color="Maroon" title="c15">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="c14">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E71">A760</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="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> <span class="kw"> such that </span><br/><a NAME="E6:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E74">A763</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E75">A764</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">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>)
 <span class="kw">and </span><br/><a NAME="E8:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E76">A765</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E9:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E77">A766</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c19">i2</font>,<font color="Maroon" title="c20">j2</font>)
 <span class="kw">and </span><br/><a NAME="E10:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E78">A767</font></span>: 
( ( <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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</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#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:1_1_1"><span class="lab"><font color="Green" title="E55">A704</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E71">A760</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/><a NAME="E11:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E79">A768</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E74">A763</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E12:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E80">A769</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E76">A765</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E13:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E81">A770</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E76">A765</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E14:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E82">A771</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E74">A763</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E15:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E83">A772</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E76">A765</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E16:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E84">A773</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E74">A763</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E17:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E85">A774</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E76">A765</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E18:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E86">A775</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E74">A763</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E19:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E87">A776</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E20:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E88">A777</font></span>: 
<font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E21:1_1_1_25_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A778</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c16">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">i19</font>, <font color="Maroon" title="c22">j19</font>, <font color="Maroon" title="c23">i29</font>, <font color="Maroon" title="c24">j29</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E23:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E90">A779</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i19</font>,<font color="Maroon" title="c22">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E24:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E91">A780</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">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="c21">i19</font>,<font color="Maroon" title="c22">j19</font>)
 <span class="kw">and </span><br/><a NAME="E25:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E92">A781</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">i29</font>,<font color="Maroon" title="c24">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E26:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E93">A782</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c23">i29</font>,<font color="Maroon" title="c24">j29</font>)
 <span class="kw">and </span><br/><a NAME="E27:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E94">A783</font></span>: 
( ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> ) or ( <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> ) or ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> ) or ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:1_1_1"><span class="lab"><font color="Green" title="E55">A704</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/><a NAME="E28:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E95">A784</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E90">A779</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E29:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E96">A785</font></span>: 
<font color="Maroon" title="c24">j29</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E92">A781</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E30:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E97">A786</font></span>: 
<font color="Maroon" title="c22">j19</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E90">A779</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E31:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E98">A787</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E92">A781</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E32:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E99">A788</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E90">A779</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E33:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E100">A789</font></span>: 
<font color="Maroon" title="c23">i29</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E92">A781</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E34:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E101">A790</font></span>: 
<font color="Maroon" title="c21">i19</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E90">A779</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><span class="kw">assume </span><a NAME="E35:1_1_1_25_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c16">j</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E36:1_1_1_25_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A791</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">i</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c16">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E37:1_1_1_25_1_1"/><span class="lab"><font color="Green" title="E103">A792</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E92">A781</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_1_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#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> &amp; <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</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="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">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> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">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> &amp; <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">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> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">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> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) 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#K2" title="NAT_1:func.2">+</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> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</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#K2" title="NAT_1:func.2">+</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> &amp; <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</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#K2" title="NAT_1:func.2">+</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> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</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#K2" title="NAT_1:func.2">+</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> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) 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#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</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#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</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#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</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#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E78">A767</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E94">A783</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_1"/><span class="lab"><font color="Green" title="E104">A793</font></span>: 
( <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#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A794</font></span>: 
<font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E12:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E80">A769</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E96">A785</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="ref" href="goboard7.html#T19" title="GOBOARD7:th.19">GOBOARD7:19</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_1_1_1_1_1_1"/>
( <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> or <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c18">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E12:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E80">A769</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E96">A785</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_1"><span class="lab"><font color="Green" title="E104">A793</font></span></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E73">A762</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_1"><span class="lab"><font color="Green" title="E105">A794</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_1"><span class="lab"><font color="Green" title="E104">A793</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_1"><span class="lab"><font color="Green" title="E105">A794</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_1_1_1_3"/>
<font color="Maroon" title="c18">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_1_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E71">A760</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_1"><span class="lab"><font color="Green" title="E104">A793</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_1"><span class="lab"><font color="Green" title="E105">A794</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_25_1_1_1_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_2"/><span class="lab"><font color="Green" title="E104">A795</font></span>: 
( <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#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> &amp; <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_1_1_1_2_1_1"/>
( ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E12:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E80">A769</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E100">A789</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_2"><span class="lab"><font color="Green" title="E104">A795</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E73">A762</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_2_1_1_2"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E87">A776</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_2"><span class="lab"><font color="Green" title="E104">A795</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_2_1_1_3"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_2_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_2"><span class="lab"><font color="Green" title="E104">A795</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_2_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_2_1_1_4"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_2_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E87">A776</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_2"><span class="lab"><font color="Green" title="E104">A795</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_25_1_1_1_1_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_3"/><span class="lab"><font color="Green" title="E104">A796</font></span>: 
( <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#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_1_1_1_3_1_1"/>
( ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E12:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E80">A769</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E37:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E103">A792</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_3"><span class="lab"><font color="Green" title="E104">A796</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_3_1_1_1"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_3_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_3"><span class="lab"><font color="Green" title="E104">A796</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_3_1_1_2"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_3_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E87">A776</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_3"><span class="lab"><font color="Green" title="E104">A796</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_3_1_1_3"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_3_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E73">A762</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_3"><span class="lab"><font color="Green" title="E104">A796</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_3_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_3_1_1_4"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_3_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E71">A760</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_3"><span class="lab"><font color="Green" title="E104">A796</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_25_1_1_1_1_3"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_4"/><span class="lab"><font color="Green" title="E104">A797</font></span>: 
( <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#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j2</font> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A798</font></span>: 
<font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E12:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E80">A769</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E98">A787</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="ref" href="goboard7.html#T19" title="GOBOARD7:th.19">GOBOARD7:19</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_1_1_1_4_1_1"/>
( <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> or <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c18">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E12:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E80">A769</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E98">A787</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_4"><span class="lab"><font color="Green" title="E104">A797</font></span></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_4_1_1_1"/>
<font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_4_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E71">A760</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_4"><span class="lab"><font color="Green" title="E104">A797</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_4"><span class="lab"><font color="Green" title="E105">A798</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_4_1_1_2"/>
<font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_4_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E73">A762</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_4"><span class="lab"><font color="Green" title="E104">A797</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_4"><span class="lab"><font color="Green" title="E105">A798</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_4_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_4_1_1_3"/>
<font color="Maroon" title="c18">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_4_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E87">A776</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_4"><span class="lab"><font color="Green" title="E104">A797</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_4"><span class="lab"><font color="Green" title="E105">A798</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_25_1_1_1_1_4"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_5"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_5"/><span class="lab"><font color="Green" title="E104">A799</font></span>: 
( <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">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> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_5_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_5_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_1_1_1_5_1_1"/>
( ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i1</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> ) or ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i1</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> ) or ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> ) or ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E85">A774</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E96">A785</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_5"><span class="lab"><font color="Green" title="E104">A799</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_5_1_1_1"/>
( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i1</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_5_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E73">A762</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_5_1_1_2"/>
( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i1</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_5_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_5"><span class="lab"><font color="Green" title="E104">A799</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_5_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_5_1_1_3"/>
( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_5_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E71">A760</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_5"><span class="lab"><font color="Green" title="E104">A799</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_5_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_5_1_1_4"/>
( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_5_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E87">A776</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_5"><span class="lab"><font color="Green" title="E104">A799</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_25_1_1_1_1_5"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_6"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_6"/><span class="lab"><font color="Green" title="E104">A800</font></span>: 
( <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">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> &amp; <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_1_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A801</font></span>: 
<font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E85">A774</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E100">A789</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="ref" href="goboard7.html#T20" title="GOBOARD7:th.20">GOBOARD7:20</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_6_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_6_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_1_1_1_6_1_1"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> or <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E85">A774</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E100">A789</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_6"><span class="lab"><font color="Green" title="E104">A800</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_6_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_6_1_1_1"/>
<font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_6_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E73">A762</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_6"><span class="lab"><font color="Green" title="E105">A801</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_6_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_6_1_1_2"/>
<font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_6_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_6"><span class="lab"><font color="Green" title="E104">A800</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_6"><span class="lab"><font color="Green" title="E105">A801</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_6_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_6_1_1_3"/>
<font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_6_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E71">A760</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_6"><span class="lab"><font color="Green" title="E104">A800</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_6"><span class="lab"><font color="Green" title="E105">A801</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_25_1_1_1_1_6"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_7"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_7"/><span class="lab"><font color="Green" title="E104">A802</font></span>: 
( <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">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> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_1_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A803</font></span>: 
<font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E85">A774</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E37:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E103">A792</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="ref" href="goboard7.html#T20" title="GOBOARD7:th.20">GOBOARD7:20</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_7_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_7_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_1_1_1_7_1_1"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> or <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E85">A774</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E37:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E103">A792</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_7"><span class="lab"><font color="Green" title="E104">A802</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_7_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_7_1_1_1"/>
<font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_7_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_7"><span class="lab"><font color="Green" title="E104">A802</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_7"><span class="lab"><font color="Green" title="E105">A803</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_7_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_7_1_1_2"/>
<font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_7_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E73">A762</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_7"><span class="lab"><font color="Green" title="E104">A802</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_7"><span class="lab"><font color="Green" title="E105">A803</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_7_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_7_1_1_3"/>
<font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_7_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E87">A776</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_7"><span class="lab"><font color="Green" title="E104">A802</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_7"><span class="lab"><font color="Green" title="E105">A803</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_25_1_1_1_1_7"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_8"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_8"/><span class="lab"><font color="Green" title="E104">A804</font></span>: 
( <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">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> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_8_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_8_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_1_1_1_8_1_1"/>
( ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i1</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> ) or ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i1</font> &amp; <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> ) or ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> ) or ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E85">A774</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E98">A787</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_8"><span class="lab"><font color="Green" title="E104">A804</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_8_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_8_1_1_1"/>
( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i1</font> &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_8_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_8"><span class="lab"><font color="Green" title="E104">A804</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_8_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_8_1_1_2"/>
( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i1</font> &amp; <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_8_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E73">A762</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_8"><span class="lab"><font color="Green" title="E104">A804</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_8_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_8_1_1_3"/>
( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_8_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E87">A776</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_8"><span class="lab"><font color="Green" title="E104">A804</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_8_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_8_1_1_4"/>
( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_8_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E71">A760</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_8"><span class="lab"><font color="Green" title="E104">A804</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_25_1_1_1_1_8"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_9"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_9"/><span class="lab"><font color="Green" title="E104">A805</font></span>: 
( <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#K2" title="NAT_1:func.2">+</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> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_9_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_9_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_1_1_1_9_1_1"/>
( ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> ) or ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> ) or ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> ) or ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E81">A770</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E96">A785</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_9"><span class="lab"><font color="Green" title="E104">A805</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_9_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_9_1_1_1"/>
( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_9_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E71">A760</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_9"><span class="lab"><font color="Green" title="E104">A805</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_9_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_9_1_1_2"/>
( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_9_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E87">A776</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_9"><span class="lab"><font color="Green" title="E104">A805</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_9_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_9_1_1_3"/>
( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_9_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E73">A762</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_9"><span class="lab"><font color="Green" title="E104">A805</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_9_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_9_1_1_4"/>
( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_9_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_9"><span class="lab"><font color="Green" title="E104">A805</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_25_1_1_1_1_9"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_10"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_10"/><span class="lab"><font color="Green" title="E104">A806</font></span>: 
( <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#K2" title="NAT_1:func.2">+</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> &amp; <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_1_1_1_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A807</font></span>: 
<font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E81">A770</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E100">A789</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="ref" href="goboard7.html#T20" title="GOBOARD7:th.20">GOBOARD7:20</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_10_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_10_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_1_1_1_10_1_1"/>
( <font color="Maroon" title="c19">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> or <font color="Maroon" title="c19">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c19">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E81">A770</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E100">A789</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_10"><span class="lab"><font color="Green" title="E104">A806</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_10_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_10_1_1_1"/>
<font color="Maroon" title="c19">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_10_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E71">A760</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_10"><span class="lab"><font color="Green" title="E104">A806</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_10"><span class="lab"><font color="Green" title="E105">A807</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_10_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_10_1_1_2"/>
<font color="Maroon" title="c19">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_10_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E87">A776</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_10"><span class="lab"><font color="Green" title="E104">A806</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_10"><span class="lab"><font color="Green" title="E105">A807</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_10_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_10_1_1_3"/>
<font color="Maroon" title="c19">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_10_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E73">A762</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_10"><span class="lab"><font color="Green" title="E104">A806</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_10"><span class="lab"><font color="Green" title="E105">A807</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_25_1_1_1_1_10"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_11"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_11"/><span class="lab"><font color="Green" title="E104">A808</font></span>: 
( <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#K2" title="NAT_1:func.2">+</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> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_1_1_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A809</font></span>: 
<font color="Maroon" title="c18">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E81">A770</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E37:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E103">A792</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="ref" href="goboard7.html#T20" title="GOBOARD7:th.20">GOBOARD7:20</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_11_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_11_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_1_1_1_11_1_1"/>
( <font color="Maroon" title="c19">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> or <font color="Maroon" title="c19">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c19">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E81">A770</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E37:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E103">A792</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_11"><span class="lab"><font color="Green" title="E104">A808</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_11_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_11_1_1_1"/>
<font color="Maroon" title="c19">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_11_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E87">A776</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_11"><span class="lab"><font color="Green" title="E104">A808</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_11"><span class="lab"><font color="Green" title="E105">A809</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_11_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_11_1_1_2"/>
<font color="Maroon" title="c19">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_11_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E71">A760</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_11"><span class="lab"><font color="Green" title="E104">A808</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_11"><span class="lab"><font color="Green" title="E105">A809</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_11_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_11_1_1_3"/>
<font color="Maroon" title="c19">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_11_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_11"><span class="lab"><font color="Green" title="E104">A808</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_11"><span class="lab"><font color="Green" title="E105">A809</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_25_1_1_1_1_11"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_12"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_12"/><span class="lab"><font color="Green" title="E104">A810</font></span>: 
( <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#K2" title="NAT_1:func.2">+</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> &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_12_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_12_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_1_1_1_12_1_1"/>
( ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c24">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> ) or ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> ) or ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> ) or ( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E86">A775</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E81">A770</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E98">A787</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_12"><span class="lab"><font color="Green" title="E104">A810</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_12_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_12_1_1_1"/>
( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c24">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_12_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E87">A776</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_12"><span class="lab"><font color="Green" title="E104">A810</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_12_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_12_1_1_2"/>
( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> &amp; <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_12_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E71">A760</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_12"><span class="lab"><font color="Green" title="E104">A810</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_12_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_12_1_1_3"/>
( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_12_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_12"><span class="lab"><font color="Green" title="E104">A810</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_12_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_12_1_1_4"/>
( <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_12_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E73">A762</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_12"><span class="lab"><font color="Green" title="E104">A810</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_25_1_1_1_1_12"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_13"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_13"/><span class="lab"><font color="Green" title="E104">A811</font></span>: 
( <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#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_1_1_1_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A812</font></span>: 
<font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E83">A772</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E96">A785</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="ref" href="goboard7.html#T19" title="GOBOARD7:th.19">GOBOARD7:19</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_13_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_13_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_1_1_1_13_1_1"/>
( <font color="Maroon" title="c20">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> or <font color="Maroon" title="c20">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c20">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E83">A772</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E96">A785</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_13"><span class="lab"><font color="Green" title="E104">A811</font></span></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_13_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_13_1_1_1"/>
<font color="Maroon" title="c20">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_13_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E71">A760</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_13"><span class="lab"><font color="Green" title="E104">A811</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_13"><span class="lab"><font color="Green" title="E105">A812</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_13_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_13_1_1_2"/>
<font color="Maroon" title="c20">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_13_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E87">A776</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_13"><span class="lab"><font color="Green" title="E104">A811</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_13"><span class="lab"><font color="Green" title="E105">A812</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_13_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_13_1_1_3"/>
<font color="Maroon" title="c20">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_13_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E73">A762</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_13"><span class="lab"><font color="Green" title="E104">A811</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_13"><span class="lab"><font color="Green" title="E105">A812</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_25_1_1_1_1_13"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_14"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_14"/><span class="lab"><font color="Green" title="E104">A813</font></span>: 
( <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#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_14_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_14_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_1_1_1_14_1_1"/>
( ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> &amp; <font color="Maroon" title="c20">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> &amp; <font color="Maroon" title="c20">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E83">A772</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E100">A789</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_14"><span class="lab"><font color="Green" title="E104">A813</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_14_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_14_1_1_1"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> &amp; <font color="Maroon" title="c20">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_14_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E71">A760</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_14"><span class="lab"><font color="Green" title="E104">A813</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_14_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_14_1_1_2"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> &amp; <font color="Maroon" title="c20">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_14_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E73">A762</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_14"><span class="lab"><font color="Green" title="E104">A813</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_14_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_14_1_1_3"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_14_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E87">A776</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_14"><span class="lab"><font color="Green" title="E104">A813</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_14_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_14_1_1_4"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_14_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_14"><span class="lab"><font color="Green" title="E104">A813</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_25_1_1_1_1_14"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_15"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_15"/><span class="lab"><font color="Green" title="E104">A814</font></span>: 
( <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#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_15_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_15_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_1_1_1_15_1_1"/>
( ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c20">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c20">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> ) or ( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E83">A772</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E99">A788</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E37:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E103">A792</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_15"><span class="lab"><font color="Green" title="E104">A814</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_15_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_15_1_1_1"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c20">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_15_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E87">A776</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_15"><span class="lab"><font color="Green" title="E104">A814</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_15_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_15_1_1_2"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c20">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_15_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_15"><span class="lab"><font color="Green" title="E104">A814</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_15_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_15_1_1_3"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_15_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E71">A760</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_15"><span class="lab"><font color="Green" title="E104">A814</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_15_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_15_1_1_4"/>
( <font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_15_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E73">A762</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_15"><span class="lab"><font color="Green" title="E104">A814</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_25_1_1_1_1_15"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_16"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_16"/><span class="lab"><font color="Green" title="E104">A815</font></span>: 
( <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#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c21">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i29</font> &amp; <font color="Maroon" title="c22">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_1_1_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A816</font></span>: 
<font color="Maroon" title="c17">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E83">A772</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E98">A787</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="ref" href="goboard7.html#T19" title="GOBOARD7:th.19">GOBOARD7:19</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_16_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_16_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_1_1_1_16_1_1"/>
( <font color="Maroon" title="c20">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> or <font color="Maroon" title="c20">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c20">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E72">A761</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E79">A768</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E82">A771</font></span></a>, <a class="txt" href="#E16:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E84">A773</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E83">A772</font></span></a>, <a class="txt" href="#E21:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E89">A778</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E28:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E95">A784</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E101">A790</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E97">A786</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E98">A787</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E102">A791</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_16"><span class="lab"><font color="Green" title="E104">A815</font></span></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_16_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_16_1_1_1"/>
<font color="Maroon" title="c20">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_16_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E87">A776</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_16"><span class="lab"><font color="Green" title="E104">A815</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_16"><span class="lab"><font color="Green" title="E105">A816</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_16_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_16_1_1_2"/>
<font color="Maroon" title="c20">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_16_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E71">A760</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E77">A766</font></span></a>, <a class="txt" href="#E24:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E91">A780</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_16"><span class="lab"><font color="Green" title="E104">A815</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_16"><span class="lab"><font color="Green" title="E105">A816</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_1_1_1_16_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_1_1_1_16_1_1_3"/>
<font color="Maroon" title="c20">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_1_1_1_16_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E70">A759</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E75">A764</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E88">A777</font></span></a>, <a class="txt" href="#E26:1_1_1_25_1_1"><span class="lab"><font color="Green" title="E93">A782</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_1_1_1_16"><span class="lab"><font color="Green" title="E104">A815</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_1_1_1_16"><span class="lab"><font color="Green" title="E105">A816</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_25_1_1_1_1_16"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E39:1_1_1_25_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_2"/>
( <font color="Maroon" title="c15">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">i</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c16">j</font>)</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_2"/><span class="kw">then </span>
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E3:1_1_1_25_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c16">j</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E70">A817</font></span>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">i</font> &amp; <font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">i</font>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c16">j</font>)</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">i19</font>, <font color="Maroon" title="c18">j19</font>, <font color="Maroon" title="c19">i29</font>, <font color="Maroon" title="c20">j29</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E71">A818</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c17">i19</font>,<font color="Maroon" title="c18">j19</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E72">A819</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">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="c17">i19</font>,<font color="Maroon" title="c18">j19</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E73">A820</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c19">i29</font>,<font color="Maroon" title="c20">j29</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E74">A821</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c19">i29</font>,<font color="Maroon" title="c20">j29</font>)
 <span class="kw">and </span><br/><a NAME="E7:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E75">A822</font></span>: 
( ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> ) or ( <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> ) or ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> ) or ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:1_1_1"><span class="lab"><font color="Green" title="E55">A704</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/><a NAME="E8:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E76">A823</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E71">A818</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E9:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E77">A824</font></span>: 
<font color="Maroon" title="c20">j29</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E73">A820</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E10:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E78">A825</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E73">A820</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E11:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E79">A826</font></span>: 
<font color="Maroon" title="c17">i19</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E71">A818</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E12:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E80">A827</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c20">j29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E73">A820</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E13:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E81">A828</font></span>: 
<font color="Maroon" title="c18">j19</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E71">A818</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E14:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E82">A829</font></span>: 
<font color="Maroon" title="c19">i29</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E73">A820</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E15:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E83">A830</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c18">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E71">A818</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><span class="kw">assume </span><a NAME="E16:1_1_1_25_1_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">i</font>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c16">j</font>)</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E17:1_1_1_25_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A831</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">i</font>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c16">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/><a NAME="E18:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E85">A832</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E19:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E86">A833</font></span>: 
<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="c14">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/><a NAME="E20:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E87">A834</font></span>: 
<font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c14">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E21:1_1_1_25_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A835</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c15">i</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/><a NAME="E22:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E89">A836</font></span>: 
<font color="Maroon" title="c15">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="c14">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E87">A834</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">i1</font>, <font color="Maroon" title="c22">j1</font>, <font color="Maroon" title="c23">i2</font>, <font color="Maroon" title="c24">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E24:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E90">A837</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c21">i1</font>,<font color="Maroon" title="c22">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E25:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E91">A838</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c15">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="c21">i1</font>,<font color="Maroon" title="c22">j1</font>)
 <span class="kw">and </span><br/><a NAME="E26:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E92">A839</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c23">i2</font>,<font color="Maroon" title="c24">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E27:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E93">A840</font></span>: 
<font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c15">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c23">i2</font>,<font color="Maroon" title="c24">j2</font>)
 <span class="kw">and </span><br/><a NAME="E28:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E94">A841</font></span>: 
( ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:1_1_1"><span class="lab"><font color="Green" title="E55">A704</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E87">A834</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/><a NAME="E29:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E95">A842</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c21">i1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E90">A837</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E30:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E96">A843</font></span>: 
<font color="Maroon" title="c24">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E92">A839</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E31:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E97">A844</font></span>: 
<font color="Maroon" title="c22">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E90">A837</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E32:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E98">A845</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c24">j2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E92">A839</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E33:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E99">A846</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">j1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E90">A837</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E34:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E100">A847</font></span>: 
<font color="Maroon" title="c23">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E92">A839</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E35:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E101">A848</font></span>: 
<font color="Maroon" title="c21">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E90">A837</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E36:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E102">A849</font></span>: 
 <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c14">g</font>,<font color="Maroon" title="c16">j</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c16">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c14">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/><a NAME="E37:1_1_1_25_1_3"/><span class="lab"><font color="Green" title="E103">A850</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">i2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E92">A839</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_3_1_1"/>
( ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E28:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E94">A841</font></span></a>, <a class="txt" href="#E7:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E75">A822</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_1"/><span class="lab"><font color="Green" title="E104">A851</font></span>: 
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A852</font></span>: 
<font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E96">A843</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E77">A824</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="ref" href="goboard7.html#T19" title="GOBOARD7:th.19">GOBOARD7:19</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_3_1_1_1_1_1"/>
( <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> or <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E96">A843</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E77">A824</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_1"><span class="lab"><font color="Green" title="E104">A851</font></span></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_1_1_1_1"/>
<font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_1_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E22:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E89">A836</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_1"><span class="lab"><font color="Green" title="E105">A852</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_1_1_1_2"/>
<font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:1_1_1"><span class="lab"><font color="Green" title="E62">A746</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_1"><span class="lab"><font color="Green" title="E104">A851</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_1"><span class="lab"><font color="Green" title="E105">A852</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_1_1_1_3"/>
<font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_1_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E87">A834</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_1"><span class="lab"><font color="Green" title="E104">A851</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_1"><span class="lab"><font color="Green" title="E105">A852</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_25_1_3_1_1_1"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_2"/><span class="lab"><font color="Green" title="E104">A853</font></span>: 
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_3_1_1_2_1_1"/>
( ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E96">A843</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E82">A829</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_2"><span class="lab"><font color="Green" title="E104">A853</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_2_1_1_1"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E22:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E89">A836</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_2_1_1_2"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E85">A832</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_2"><span class="lab"><font color="Green" title="E104">A853</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_2_1_1_3"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_2_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:1_1_1"><span class="lab"><font color="Green" title="E62">A746</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_2"><span class="lab"><font color="Green" title="E104">A853</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_2_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_2_1_1_4"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_2_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E85">A832</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_2"><span class="lab"><font color="Green" title="E104">A853</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_25_1_3_1_1_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_3"/><span class="lab"><font color="Green" title="E104">A854</font></span>: 
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_3_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_3_1_1_3_1_1"/>
( ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E96">A843</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E10:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E78">A825</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_3"><span class="lab"><font color="Green" title="E104">A854</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_3_1_1_1"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_3_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:1_1_1"><span class="lab"><font color="Green" title="E62">A746</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_3"><span class="lab"><font color="Green" title="E104">A854</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_3_1_1_2"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_3_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E85">A832</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_3"><span class="lab"><font color="Green" title="E104">A854</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_3_1_1_3"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_3_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E22:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E89">A836</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_3"><span class="lab"><font color="Green" title="E104">A854</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_3_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_3_1_1_4"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_3_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E87">A834</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_3"><span class="lab"><font color="Green" title="E104">A854</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_25_1_3_1_1_3"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_4"/><span class="lab"><font color="Green" title="E104">A855</font></span>: 
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_3_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A856</font></span>: 
<font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E96">A843</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E12:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E80">A827</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="ref" href="goboard7.html#T19" title="GOBOARD7:th.19">GOBOARD7:19</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_4_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_3_1_1_4_1_1"/>
( <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> or <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E30:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E96">A843</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E12:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E80">A827</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_4"><span class="lab"><font color="Green" title="E104">A855</font></span></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_4_1_1_1"/>
<font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_4_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E87">A834</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_4"><span class="lab"><font color="Green" title="E104">A855</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_4"><span class="lab"><font color="Green" title="E105">A856</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_4_1_1_2"/>
<font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_4_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E22:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E89">A836</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_4"><span class="lab"><font color="Green" title="E104">A855</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_4"><span class="lab"><font color="Green" title="E105">A856</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_4_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_4_1_1_3"/>
<font color="Maroon" title="c22">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_4_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E85">A832</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_4"><span class="lab"><font color="Green" title="E104">A855</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_4"><span class="lab"><font color="Green" title="E105">A856</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_25_1_3_1_1_4"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_5"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_5"/><span class="lab"><font color="Green" title="E104">A857</font></span>: 
( <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_5_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_5_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_3_1_1_5_1_1"/>
( ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i1</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> ) or ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i1</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> ) or ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> ) or ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E100">A847</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E77">A824</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_5"><span class="lab"><font color="Green" title="E104">A857</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_5_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_5_1_1_1"/>
( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i1</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_5_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E22:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E89">A836</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_5_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_5_1_1_2"/>
( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i1</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_5_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:1_1_1"><span class="lab"><font color="Green" title="E62">A746</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_5"><span class="lab"><font color="Green" title="E104">A857</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_5_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_5_1_1_3"/>
( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_5_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E87">A834</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_5"><span class="lab"><font color="Green" title="E104">A857</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_5_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_5_1_1_4"/>
( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_5_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E85">A832</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_5"><span class="lab"><font color="Green" title="E104">A857</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_25_1_3_1_1_5"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_6"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_6"/><span class="lab"><font color="Green" title="E104">A858</font></span>: 
( <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_3_1_1_6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A859</font></span>: 
<font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E100">A847</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E82">A829</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="ref" href="goboard7.html#T20" title="GOBOARD7:th.20">GOBOARD7:20</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_6_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_6_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_3_1_1_6_1_1"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> or <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E100">A847</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E82">A829</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_6"><span class="lab"><font color="Green" title="E104">A858</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_6_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_6_1_1_1"/>
<font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_6_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E22:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E89">A836</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_6"><span class="lab"><font color="Green" title="E105">A859</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_6_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_6_1_1_2"/>
<font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_6_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:1_1_1"><span class="lab"><font color="Green" title="E62">A746</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_6"><span class="lab"><font color="Green" title="E104">A858</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_6"><span class="lab"><font color="Green" title="E105">A859</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_6_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_6_1_1_3"/>
<font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_6_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E87">A834</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_6"><span class="lab"><font color="Green" title="E104">A858</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_6"><span class="lab"><font color="Green" title="E105">A859</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_25_1_3_1_1_6"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_7"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_7"/><span class="lab"><font color="Green" title="E104">A860</font></span>: 
( <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_3_1_1_7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A861</font></span>: 
<font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E100">A847</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E10:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E78">A825</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="ref" href="goboard7.html#T20" title="GOBOARD7:th.20">GOBOARD7:20</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_7_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_7_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_3_1_1_7_1_1"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> or <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E100">A847</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E10:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E78">A825</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_7"><span class="lab"><font color="Green" title="E104">A860</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_7_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_7_1_1_1"/>
<font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_7_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:1_1_1"><span class="lab"><font color="Green" title="E62">A746</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_7"><span class="lab"><font color="Green" title="E104">A860</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_7"><span class="lab"><font color="Green" title="E105">A861</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_7_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_7_1_1_2"/>
<font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_7_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E22:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E89">A836</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_7"><span class="lab"><font color="Green" title="E104">A860</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_7"><span class="lab"><font color="Green" title="E105">A861</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_7_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_7_1_1_3"/>
<font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_7_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E85">A832</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_7"><span class="lab"><font color="Green" title="E104">A860</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_7"><span class="lab"><font color="Green" title="E105">A861</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_25_1_3_1_1_7"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_8"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_8"/><span class="lab"><font color="Green" title="E104">A862</font></span>: 
( <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_8_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_8_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_3_1_1_8_1_1"/>
( ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i1</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> ) or ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i1</font> &amp; <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> ) or ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> ) or ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E34:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E100">A847</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E12:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E80">A827</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_8"><span class="lab"><font color="Green" title="E104">A862</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_8_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_8_1_1_1"/>
( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i1</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_8_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:1_1_1"><span class="lab"><font color="Green" title="E62">A746</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_8"><span class="lab"><font color="Green" title="E104">A862</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_8_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_8_1_1_2"/>
( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i1</font> &amp; <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_8_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E22:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E89">A836</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_8"><span class="lab"><font color="Green" title="E104">A862</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_8_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_8_1_1_3"/>
( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_8_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E85">A832</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_8"><span class="lab"><font color="Green" title="E104">A862</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_8_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_8_1_1_4"/>
( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_8_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E87">A834</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_8"><span class="lab"><font color="Green" title="E104">A862</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_25_1_3_1_1_8"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_9"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_9"/><span class="lab"><font color="Green" title="E104">A863</font></span>: 
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_9_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_9_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_3_1_1_9_1_1"/>
( ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> ) or ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> ) or ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> ) or ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E37:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E103">A850</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E77">A824</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_9"><span class="lab"><font color="Green" title="E104">A863</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_9_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_9_1_1_1"/>
( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_9_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E87">A834</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_9"><span class="lab"><font color="Green" title="E104">A863</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_9_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_9_1_1_2"/>
( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_9_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E85">A832</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_9"><span class="lab"><font color="Green" title="E104">A863</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_9_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_9_1_1_3"/>
( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_9_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E22:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E89">A836</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_9"><span class="lab"><font color="Green" title="E104">A863</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_9_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_9_1_1_4"/>
( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_9_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:1_1_1"><span class="lab"><font color="Green" title="E62">A746</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_9"><span class="lab"><font color="Green" title="E104">A863</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_25_1_3_1_1_9"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_10"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_10"/><span class="lab"><font color="Green" title="E104">A864</font></span>: 
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_3_1_1_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A865</font></span>: 
<font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E37:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E103">A850</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E82">A829</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="ref" href="goboard7.html#T20" title="GOBOARD7:th.20">GOBOARD7:20</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_10_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_10_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_3_1_1_10_1_1"/>
( <font color="Maroon" title="c23">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> or <font color="Maroon" title="c23">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E37:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E103">A850</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E82">A829</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_10"><span class="lab"><font color="Green" title="E104">A864</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_10_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_10_1_1_1"/>
<font color="Maroon" title="c23">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_10_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E87">A834</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_10"><span class="lab"><font color="Green" title="E104">A864</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_10"><span class="lab"><font color="Green" title="E105">A865</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_10_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_10_1_1_2"/>
<font color="Maroon" title="c23">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_10_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E85">A832</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_10"><span class="lab"><font color="Green" title="E104">A864</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_10"><span class="lab"><font color="Green" title="E105">A865</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_10_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_10_1_1_3"/>
<font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_10_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E22:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E89">A836</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_10"><span class="lab"><font color="Green" title="E104">A864</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_10"><span class="lab"><font color="Green" title="E105">A865</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_25_1_3_1_1_10"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_11"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_11"/><span class="lab"><font color="Green" title="E104">A866</font></span>: 
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_3_1_1_11"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A867</font></span>: 
<font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E37:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E103">A850</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E10:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E78">A825</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="ref" href="goboard7.html#T20" title="GOBOARD7:th.20">GOBOARD7:20</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_11_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_11_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_3_1_1_11_1_1"/>
( <font color="Maroon" title="c23">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> or <font color="Maroon" title="c23">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E37:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E103">A850</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E10:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E78">A825</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_11"><span class="lab"><font color="Green" title="E104">A866</font></span></a>, <a class="ref" href="goboard7.html#T23" title="GOBOARD7:th.23">GOBOARD7:23</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_11_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_11_1_1_1"/>
<font color="Maroon" title="c23">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_11_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E85">A832</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_11"><span class="lab"><font color="Green" title="E104">A866</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_11"><span class="lab"><font color="Green" title="E105">A867</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_11_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_11_1_1_2"/>
<font color="Maroon" title="c23">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_11_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E87">A834</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_11"><span class="lab"><font color="Green" title="E104">A866</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_11"><span class="lab"><font color="Green" title="E105">A867</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_11_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_11_1_1_3"/>
<font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_11_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:1_1_1"><span class="lab"><font color="Green" title="E62">A746</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_11"><span class="lab"><font color="Green" title="E104">A866</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_11"><span class="lab"><font color="Green" title="E105">A867</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_25_1_3_1_1_11"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_12"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_12"/><span class="lab"><font color="Green" title="E104">A868</font></span>: 
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_12_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_12_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_3_1_1_12_1_1"/>
( ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c20">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> ) or ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> ) or ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> ) or ( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E33:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E99">A846</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E37:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E103">A850</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E12:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E80">A827</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_12"><span class="lab"><font color="Green" title="E104">A868</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_12_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_12_1_1_1"/>
( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c20">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_12_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E85">A832</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_12"><span class="lab"><font color="Green" title="E104">A868</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_12_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_12_1_1_2"/>
( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_12_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E87">A834</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_12"><span class="lab"><font color="Green" title="E104">A868</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_12_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_12_1_1_3"/>
( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j29</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_12_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:1_1_1"><span class="lab"><font color="Green" title="E62">A746</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_12"><span class="lab"><font color="Green" title="E104">A868</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_12_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_12_1_1_4"/>
( <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">j1</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_12_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E22:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E89">A836</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_12"><span class="lab"><font color="Green" title="E104">A868</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_25_1_3_1_1_12"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_13"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_13"/><span class="lab"><font color="Green" title="E104">A869</font></span>: 
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_3_1_1_13"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A870</font></span>: 
<font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E98">A845</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E77">A824</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="ref" href="goboard7.html#T19" title="GOBOARD7:th.19">GOBOARD7:19</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_13_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_13_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_3_1_1_13_1_1"/>
( <font color="Maroon" title="c24">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> or <font color="Maroon" title="c24">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E98">A845</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E9:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E77">A824</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_13"><span class="lab"><font color="Green" title="E104">A869</font></span></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_13_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_13_1_1_1"/>
<font color="Maroon" title="c24">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_13_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E87">A834</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_13"><span class="lab"><font color="Green" title="E104">A869</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_13"><span class="lab"><font color="Green" title="E105">A870</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_13_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_13_1_1_2"/>
<font color="Maroon" title="c24">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_13_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E85">A832</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_13"><span class="lab"><font color="Green" title="E104">A869</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_13"><span class="lab"><font color="Green" title="E105">A870</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_13_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_13_1_1_3"/>
<font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_13_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E22:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E89">A836</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_13"><span class="lab"><font color="Green" title="E104">A869</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_13"><span class="lab"><font color="Green" title="E105">A870</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_25_1_3_1_1_13"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_14"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_14"/><span class="lab"><font color="Green" title="E104">A871</font></span>: 
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_14_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_14_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_3_1_1_14_1_1"/>
( ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> &amp; <font color="Maroon" title="c24">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> &amp; <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E98">A845</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E14:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E82">A829</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_14"><span class="lab"><font color="Green" title="E104">A871</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_14_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_14_1_1_1"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> &amp; <font color="Maroon" title="c24">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_14_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E87">A834</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_14"><span class="lab"><font color="Green" title="E104">A871</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_14_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_14_1_1_2"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> &amp; <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_14_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E22:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E89">A836</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_14"><span class="lab"><font color="Green" title="E104">A871</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_14_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_14_1_1_3"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_14_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E85">A832</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_14"><span class="lab"><font color="Green" title="E104">A871</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_14_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_14_1_1_4"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_14_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:1_1_1"><span class="lab"><font color="Green" title="E62">A746</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_14"><span class="lab"><font color="Green" title="E104">A871</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_25_1_3_1_1_14"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_15"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_15"/><span class="lab"><font color="Green" title="E104">A872</font></span>: 
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_15_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_15_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_3_1_1_15_1_1"/>
( ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c24">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> ) or ( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E98">A845</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E15:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E83">A830</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E10:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E78">A825</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_15"><span class="lab"><font color="Green" title="E104">A872</font></span></a>, <a class="ref" href="goboard7.html#T21" title="GOBOARD7:th.21">GOBOARD7:21</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_15_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_15_1_1_1"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c24">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_15_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E85">A832</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_15"><span class="lab"><font color="Green" title="E104">A872</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_15_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_15_1_1_2"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_15_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:1_1_1"><span class="lab"><font color="Green" title="E62">A746</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_15"><span class="lab"><font color="Green" title="E104">A872</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_15_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_15_1_1_3"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_15_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E87">A834</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_15"><span class="lab"><font color="Green" title="E104">A872</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_15_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_15_1_1_4"/>
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">j19</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_15_1_1_4"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E5:1_1_1_25"><span class="lab"><font color="Green" title="E67">A756</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E22:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E89">A836</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_15"><span class="lab"><font color="Green" title="E104">A872</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_25_1_3_1_1_15"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_16"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_16"/><span class="lab"><font color="Green" title="E104">A873</font></span>: 
( <font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">i2</font> &amp; <font color="Maroon" title="c22">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c17">i19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">i29</font> &amp; <font color="Maroon" title="c18">j19</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_25_1_3_1_1_16"/><span class="kw">then </span><span class="lab"><font color="Green" title="E105">A874</font></span>: 
<font color="Maroon" title="c21">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">i19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E98">A845</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E12:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E80">A827</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="ref" href="goboard7.html#T19" title="GOBOARD7:th.19">GOBOARD7:19</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_16_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_16_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_25_1_3_1_1_16_1_1"/>
( <font color="Maroon" title="c24">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> or <font color="Maroon" title="c24">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E88">A835</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E29:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E95">A842</font></span></a>, <a class="txt" href="#E35:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E101">A848</font></span></a>, <a class="txt" href="#E31:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E97">A844</font></span></a>, <a class="txt" href="#E32:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E98">A845</font></span></a>, <a class="txt" href="#E36:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E102">A849</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E8:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E76">A823</font></span></a>, <a class="txt" href="#E11:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E79">A826</font></span></a>, <a class="txt" href="#E13:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E81">A828</font></span></a>, <a class="txt" href="#E12:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E80">A827</font></span></a>, <a class="txt" href="#E17:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E84">A831</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_16"><span class="lab"><font color="Green" title="E104">A873</font></span></a>, <a class="ref" href="goboard7.html#T22" title="GOBOARD7:th.22">GOBOARD7:22</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_16_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_16_1_1_1"/>
<font color="Maroon" title="c24">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_16_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E78:1_1_1"><span class="lab"><font color="Green" title="E61">A738</font></span></a>, <a class="txt" href="#E6:1_1_1_25"><span class="lab"><font color="Green" title="E68">A757</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E18:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E85">A832</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_16"><span class="lab"><font color="Green" title="E104">A873</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_16"><span class="lab"><font color="Green" title="E105">A874</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_16_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_16_1_1_2"/>
<font color="Maroon" title="c24">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_16_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E75:1_1_1"><span class="lab"><font color="Green" title="E59">A710</font></span></a>, <a class="txt" href="#E1:1_1_1_25"><span class="lab"><font color="Green" title="E63">A752</font></span></a>, <a class="txt" href="#E4:1_1_1_25"><span class="lab"><font color="Green" title="E66">A755</font></span></a>, <a class="txt" href="#E3:1_1_1_25"><span class="lab"><font color="Green" title="E65">A754</font></span></a>, <a class="txt" href="#E20:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E87">A834</font></span></a>, <a class="txt" href="#E19:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E86">A833</font></span></a>, <a class="txt" href="#E27:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E93">A840</font></span></a>, <a class="txt" href="#E4:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E72">A819</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_16"><span class="lab"><font color="Green" title="E104">A873</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_16"><span class="lab"><font color="Green" title="E105">A874</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_25_1_3_1_1_16_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_25_1_3_1_1_16_1_1_3"/>
<font color="Maroon" title="c24">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">j29</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:1_1_1_25_1_3_1_1_16_1_1_3"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E79:1_1_1"><span class="lab"><font color="Green" title="E62">A746</font></span></a>, <a class="txt" href="#E8:1_1_1_25"><span class="lab"><font color="Green" title="E69">A758</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E70">A817</font></span></a>, <a class="txt" href="#E25:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E91">A838</font></span></a>, <a class="txt" href="#E6:1_1_1_25_1_3"><span class="lab"><font color="Green" title="E74">A821</font></span></a>, <a class="txt" href="#E1:1_1_1_25_1_3_1_1_16"><span class="lab"><font color="Green" title="E104">A873</font></span></a>, <a class="txt" href="#E2:1_1_1_25_1_3_1_1_16"><span class="lab"><font color="Green" title="E105">A874</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E4:1_1_1_25_1_3_1_1_16"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E39:1_1_1_25_1_3"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c15">g</font> = <font color="Maroon" title="c14">g</font> as   non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:1_1_1"><span class="lab"><font color="Green" title="E39">A650</font></span></a>, <a class="txt" href="#E77:1_1_1"><span class="lab"><font color="Green" title="E60">A737</font></span></a>, <a class="txt" href="#E67:1_1_1"><span class="lab"><font color="Green" title="E52">A701</font></span></a>, <a class="txt" href="#E71:1_1_1"><span class="lab"><font color="Green" title="E55">A704</font></span></a>, <a class="txt" href="#E73:1_1_1"><span class="lab"><font color="Green" title="E57">A706</font></span></a>, <a class="ref" href="finseq_6.html#D1" title="FINSEQ_6:def.1">FINSEQ_6:def 1</a>, <a class="ref" href="jordan8.html#T4" title="JORDAN8:th.4">JORDAN8:4</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">Lg9</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c15">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="E83:1_1_1"/><span class="lab"><font color="Green" title="E63">A875</font></span>: 
<font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c16">Lg9</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_26"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> or <font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Lg9</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_26"/><span class="lab"><font color="Green" title="E64">A876</font></span>: 
<font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_26"/><span class="lab"><font color="Green" title="E65">A877</font></span>: 
 not <font color="Maroon" title="c17">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c16">Lg9</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c18">c</font> = <font color="Maroon" title="c17">c</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_26"><span class="lab"><font color="Green" title="E64">A876</font></span></a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c19">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E5:1_1_1_26"/><span class="lab"><font color="Green" title="E66">A878</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font>
 <span class="kw">and </span><br/><a NAME="E6:1_1_1_26"/><span class="lab"><font color="Green" title="E67">A879</font></span>: 
<font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font>
 <span class="kw">and </span><br/><a NAME="E7:1_1_1_26"/><span class="lab"><font color="Green" title="E68">A880</font></span>: 
<font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="rltopsp1.html#K1" title="RLTOPSP1:func.1">LSeg</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c15">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_26"><span class="lab"><font color="Green" title="E65">A877</font></span></a>, <a class="ref" href="sppol_2.html#T14" title="SPPOL_2:th.14">SPPOL_2:14</a>, <a class="ref" href="subset_1.html#T29" title="SUBSET_1:th.29">SUBSET_1:29</a></span>;<br/>
<a NAME="E8:1_1_1_26"/><span class="lab"><font color="Green" title="E69">A881</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_26"><span class="lab"><font color="Green" title="E66">A878</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<a NAME="E9:1_1_1_26"/>
<font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_26"><span class="lab"><font color="Green" title="E66">A878</font></span></a>, <a class="txt" href="#E6:1_1_1_26"><span class="lab"><font color="Green" title="E67">A879</font></span></a>, <a class="ref" href="seq_4.html#T134" title="SEQ_4:th.134">SEQ_4:134</a></span>;<br/>
<a NAME="E10:1_1_1_26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A882</font></span>: 
<font color="Maroon" title="c15">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T27" title="FINSEQ_5:th.27">FINSEQ_5:27</a></span>;<br/>
<a NAME="E11:1_1_1_26"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
;<br/>
<a NAME="E12:1_1_1_26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A883</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:1_1_1_26"><span class="lab"><font color="Green" title="E67">A879</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E13:1_1_1_26"/>
<font color="Maroon" title="c19">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_26"><span class="lab"><font color="Green" title="E66">A878</font></span></a>, <a class="txt" href="#E6:1_1_1_26"><span class="lab"><font color="Green" title="E67">A879</font></span></a>, <a class="ref" href="seq_4.html#T134" title="SEQ_4:th.134">SEQ_4:134</a></span>;<br/>
<a NAME="E14:1_1_1_26"/><span class="kw">then </span>
<font color="Maroon" title="c15">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T27" title="FINSEQ_5:th.27">FINSEQ_5:27</a></span>;<br/>
<a NAME="E15:1_1_1_26"/><span class="kw">then </span>
<font color="Maroon" title="c18">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E63:1_1_1"><span class="lab"><font color="Green" title="E51">A700</font></span></a>, <a class="txt" href="#E7:1_1_1_26"><span class="lab"><font color="Green" title="E68">A880</font></span></a>, <a class="txt" href="#E10:1_1_1_26"><span class="lab"><font color="Green" title="E70">A882</font></span></a>, <a class="txt" href="#E8:1_1_1_26"><span class="lab"><font color="Green" title="E69">A881</font></span></a>, <a class="txt" href="#E12:1_1_1_26"><span class="lab"><font color="Green" title="E71">A883</font></span></a>, <a class="ref" href="topreal1.html#D3" title="TOPREAL1:def.3">TOPREAL1:def 3</a></span>;<br/>
<a NAME="E16:1_1_1_26"/><span class="kw">then </span>
<font color="Maroon" title="c18">c</font> <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> (<font color="Maroon" title="c12">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E63:1_1_1"><span class="lab"><font color="Green" title="E51">A700</font></span></a>, <a class="txt" href="#E8:1_1_1_26"><span class="lab"><font color="Green" title="E69">A881</font></span></a>, <a class="txt" href="#E12:1_1_1_26"><span class="lab"><font color="Green" title="E71">A883</font></span></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></span>;<br/>
<a NAME="E17:1_1_1_26"/><span class="kw">then </span>
<font color="Maroon" title="c18">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> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E18:1_1_1_26"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c19">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_26"><span class="lab"><font color="Green" title="E64">A876</font></span></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:1_1_1_26"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E63:1_1_1"><span class="lab"><font color="Green" title="E51">A700</font></span></a>, <a class="txt" href="#E8:1_1_1_26"><span class="lab"><font color="Green" title="E69">A881</font></span></a>, <a class="txt" href="#E12:1_1_1_26"><span class="lab"><font color="Green" title="E71">A883</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E84:1_1_1"/><span class="lab"><font color="Green" title="E64">A884</font></span>: 
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c15">g</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a></span>;<br/>
<a NAME="E85:1_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T3" title="TOPS_1:th.3">TOPS_1:3</a></span>;<br/>
<a NAME="E86:1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A885</font></span>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T23" title="TOPS_1:th.23">TOPS_1:23</a></span>;<br/>
<a NAME="E87:1_1_1"/><span class="lab"><font color="Green" title="E66">A886</font></span>: 
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c15">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_27"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:1_1_1_27"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c10">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E46:1_1_1"><span class="lab"><font color="Green" title="E40">A651</font></span></a>, <a class="txt" href="#E57:1_1_1"><span class="lab"><font color="Green" title="E47">A658</font></span></a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:1_1_1_27"/><span class="lab"><font color="Green" title="E67">A887</font></span>: 
<font color="Maroon" title="c17">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="c12">f</font>,<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>)
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_27"/><span class="lab"><font color="Green" title="E68">A888</font></span>: 
<font color="Maroon" title="c17">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">p</font> = <font color="Maroon" title="c17">p</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_27"><span class="lab"><font color="Green" title="E67">A887</font></span></a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_27_1"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c19">u</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#K14" title="EUCLID:func.14">Euclid</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal3.html#T8" title="TOPREAL3:th.8">TOPREAL3:8</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c20">p</font> = <font color="Maroon" title="c18">p</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font> &amp; <font color="Maroon" title="c20">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c15">g</font> )</span><br/><span class="kw">thus </span><a NAME="E2:1_1_1_27_1"/>
<font color="Maroon" title="c20">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_27"><span class="lab"><font color="Green" title="E68">A888</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c20">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c15">g</font></span><br/><a NAME="E3:1_1_1_27_1"/><span class="lab"><font color="Green" title="E69">A889</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Maroon" title="c15">g</font>,1)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c15">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:1_1_1"><span class="lab"><font color="Green" title="E54">A703</font></span></a>, <a class="ref" href="goboard9.html#T21" title="GOBOARD9:th.21">GOBOARD9:21</a></span>;<br/><a NAME="E4:1_1_1_27_1"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c15">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#K5" title="GOBOARD5:func.5">left_cell</a> (<font color="Maroon" title="c15">g</font>,1)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:1_1_1"><span class="lab"><font color="Green" title="E55">A704</font></span></a>, <a class="txt" href="#E70:1_1_1"><span class="lab"><font color="Green" title="E54">A703</font></span></a>, <a class="ref" href="gobrd13.html#T33" title="GOBRD13:th.33">GOBRD13:33</a>, <a class="ref" href="tops_1.html#T19" title="TOPS_1:th.19">TOPS_1:19</a></span>;<br/><a NAME="E5:1_1_1_27_1"/><span class="kw">then </span>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c15">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#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c15">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_27_1"><span class="lab"><font color="Green" title="E69">A889</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E6:1_1_1_27_1"/><span class="kw">then </span>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c15">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E62:1_1_1"><span class="lab"><font color="Green" title="E50">A699</font></span></a>, <a class="txt" href="#E70:1_1_1"><span class="lab"><font color="Green" title="E54">A703</font></span></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></span>;<br/><a NAME="E7:1_1_1_27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A890</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c10">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#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c15">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:1_1_1"><span class="lab"><font color="Green" title="E40">A651</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c21">r</font> being   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> <span class="kw"> such that </span><br/><a NAME="E9:1_1_1_27_1"/><span class="lab"><font color="Green" title="E71">A891</font></span>: 
<font color="Maroon" title="c21">r</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">and </span><br/><a NAME="E10:1_1_1_27_1"/><span class="lab"><font color="Green" title="E72">A892</font></span>: 
 <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c21">r</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:1_1_1"><span class="lab"><font color="Green" title="E63">A875</font></span></a>, <a class="txt" href="#E86:1_1_1"><span class="lab"><font color="Green" title="E65">A885</font></span></a>, <a class="txt" href="#E4:1_1_1_27"><span class="lab"><font color="Green" title="E68">A888</font></span></a>, <a class="ref" href="goboard6.html#T5" title="GOBOARD6:th.5">GOBOARD6:5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">r</font> = <font color="Maroon" title="c21">r</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c23">B</font> =  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c22">r</font>) as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1"><span class="lab"><font color="Green" title="E4">A4</font></span></a>, <a class="txt" href="#E9:1_1_1_27_1"><span class="lab"><font color="Green" title="E71">A891</font></span></a>, <a class="ref" href="tbsp_1.html#T11" title="TBSP_1:th.11">TBSP_1:11</a>, <a class="ref" href="topmetr.html#T12" title="TOPMETR:th.12">TOPMETR:12</a></span>;<br/><a NAME="E13:1_1_1_27_1"/><span class="lab"><font color="Green" title="E73">A893</font></span>: 
<font color="Maroon" title="c23">B</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard6.html#T3" title="GOBOARD6:th.3">GOBOARD6:3</a></span>;<br/><a NAME="E14:1_1_1_27_1"/><span class="lab"><font color="Green" title="E74">A894</font></span>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<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="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#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c10">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:1_1_1"><span class="lab"><font color="Green" title="E40">A651</font></span></a>, <a class="txt" href="#E57:1_1_1"><span class="lab"><font color="Green" title="E47">A658</font></span></a>, <a class="txt" href="#E59:1_1_1"><span class="lab"><font color="Green" title="E49">A679</font></span></a></span>;<br/><a NAME="E15:1_1_1_27_1"/>
<font color="Maroon" title="c20">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="metric_1.html#K9" title="METRIC_1:func.9">Ball</a> (<font color="Maroon" title="c19">u</font>,<font color="Maroon" title="c22">r</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_27_1"><span class="lab"><font color="Green" title="E71">A891</font></span></a>, <a class="ref" href="tbsp_1.html#T11" title="TBSP_1:th.11">TBSP_1:11</a></span>;<br/><a NAME="E16:1_1_1_27_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A895</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c10">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="c23">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_27"><span class="lab"><font color="Green" title="E67">A887</font></span></a>, <a class="txt" href="#E14:1_1_1_27_1"><span class="lab"><font color="Green" title="E74">A894</font></span></a>, <a class="txt" href="#E13:1_1_1_27_1"><span class="lab"><font color="Green" title="E73">A893</font></span></a>, <a class="ref" href="tops_1.html#T12" title="TOPS_1:th.12">TOPS_1:12</a></span>;<br/><a NAME="E17:1_1_1_27_1"/><span class="lab"><font color="Green" title="E76">A896</font></span>: 
<font color="Maroon" title="c20">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c23">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:1_1_1_27_1"><span class="lab"><font color="Green" title="E71">A891</font></span></a>, <a class="ref" href="tbsp_1.html#T11" title="TBSP_1:th.11">TBSP_1:11</a></span>;<br/><a NAME="E18:1_1_1_27_1"/>
<font color="Maroon" title="c23">B</font> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="sprect_3.html#T7" title="SPRECT_3:th.7">SPRECT_3:7</a></span>;<br/><a NAME="E19:1_1_1_27_1"/><span class="kw">then </span>
<font color="Maroon" title="c23">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c15">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E84:1_1_1"><span class="lab"><font color="Green" title="E64">A884</font></span></a>, <a class="txt" href="#E10:1_1_1_27_1"><span class="lab"><font color="Green" title="E72">A892</font></span></a>, <a class="txt" href="#E7:1_1_1_27_1"><span class="lab"><font color="Green" title="E70">A890</font></span></a>, <a class="txt" href="#E16:1_1_1_27_1"><span class="lab"><font color="Green" title="E75">A895</font></span></a>, <a class="ref" href="goboard9.html#T4" title="GOBOARD9:th.4">GOBOARD9:4</a></span>;<br/><span class="kw">hence </span><a NAME="E20:1_1_1_27_1"/>
<font color="Maroon" title="c20">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c15">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:1_1_1_27_1"><span class="lab"><font color="Green" title="E76">A896</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E7:1_1_1_27"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c15">g</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E88:1_1_1"/><span class="lab"><font color="Green" title="E67">A897</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c15">g</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c12">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan3.html#T40" title="JORDAN3:th.40">JORDAN3:40</a></span>;<br/>
<a NAME="E89:1_1_1"/><span class="lab"><font color="Green" title="E68">A898</font></span>: 
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c15">g</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#D2" title="GOBOARD9:def.2">GOBOARD9:def 2</a></span>;<br/>
<a NAME="E90:1_1_1"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:1_1_1_28"/><span class="lab"><font color="Green" title="E69">A899</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> holds <br/><span class="p1">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c17">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">n</font> implies <span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )</span><br/>

<span class="kw">assume </span><a NAME="E1:1_1_1_28_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">n</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span><br/>

<span class="kw">hence </span><span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T38" title="NAT_D:th.38">NAT_D:38</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T37" title="NAT_D:th.37">NAT_D:37</a></span>
<br/>.= 
<font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1

;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">assume </span><a NAME="E2:1_1_1_28"/>
<font color="Maroon" title="c10">m</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E3:1_1_1_28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A900</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:1_1_1"><span class="lab"><font color="Green" title="E40">A651</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E4:1_1_1_28"/><span class="lab"><font color="Green" title="E71">A901</font></span>: 
 for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">n</font> &amp; <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 holds <br/> not <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c15">g</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:1_1_1_28_2"/><span class="lab"><font color="Green" title="E72">A902</font></span>: 
2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1"><span class="lab"><font color="Green" title="E2">A2</font></span></a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c17">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">n</font> &amp; <font color="Maroon" title="c17">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 implies  not <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c15">g</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:1_1_1_28_2"/><span class="lab"><font color="Green" title="E73">A903</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">n</font>
 <span class="kw">and </span><br/><a NAME="E3:1_1_1_28_2"/><span class="lab"><font color="Green" title="E74">A904</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c15">g</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c18">p</font> = <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font>;<br/>
<a NAME="E4:1_1_1_28_2"/><span class="lab"><font color="Green" title="E75">A905</font></span>: 
<font color="Maroon" title="c17">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="c12">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:1_1_1"><span class="lab"><font color="Green" title="E50">A699</font></span></a>, <a class="txt" href="#E3:1_1_1_28_2"><span class="lab"><font color="Green" title="E74">A904</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E5:1_1_1_28_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A906</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan1h.html#K2" title="JORDAN1H:func.2">Values</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E2:1_1_1_28_2"><span class="lab"><font color="Green" title="E73">A903</font></span></a>, <a class="ref" href="jordan9.html#T6" title="JORDAN9:th.6">JORDAN9:6</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:1_1_1_28_2"/>
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c15">g</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E8:1_1_1_28_2"/><span class="lab"><font color="Green" title="E77">A907</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">j</font>
 <span class="kw">and </span><br/><a NAME="E9:1_1_1_28_2"/><span class="lab"><font color="Green" title="E78">A908</font></span>: 
<font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">and </span><br/><a NAME="E10:1_1_1_28_2"/><span class="lab"><font color="Green" title="E79">A909</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">LSeg</a> (<font color="Maroon" title="c12">f</font>,<font color="Maroon" title="c19">j</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E62:1_1_1"><span class="lab"><font color="Green" title="E50">A699</font></span></a>, <a class="ref" href="jordan9.html#T7" title="JORDAN9:th.7">JORDAN9:7</a></span>;<br/>
<a NAME="E11:1_1_1_28_2"/><span class="lab"><font color="Green" title="E80">A910</font></span>: 
<font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a>, <a class="txt" href="#E9:1_1_1_28_2"><span class="lab"><font color="Green" title="E78">A908</font></span></a></span>;<br/>
<a NAME="E12:1_1_1_28_2"/><span class="lab"><font color="Green" title="E81">A911</font></span>: 
<font color="Maroon" title="c19">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E9:1_1_1_28_2"><span class="lab"><font color="Green" title="E78">A908</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E13:1_1_1_28_2"/><span class="lab"><font color="Green" title="E82">A912</font></span>: 
<font color="Maroon" title="c17">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_28_2"><span class="lab"><font color="Green" title="E74">A904</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E14:1_1_1_28_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A913</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_28_2"><span class="lab"><font color="Green" title="E77">A907</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E15:1_1_1_28_2"/><span class="lab"><font color="Green" title="E84">A914</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:1_1_1"><span class="lab"><font color="Green" title="E40">A651</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<a NAME="E16:1_1_1_28_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A915</font></span>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">j</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_28"><span class="lab"><font color="Green" title="E70">A900</font></span></a>, <a class="txt" href="#E8:1_1_1_28_2"><span class="lab"><font color="Green" title="E77">A907</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_28_2_1"/>
( <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">j</font> or <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E9:1_1_1_28_2"><span class="lab"><font color="Green" title="E78">A908</font></span></a>, <a class="txt" href="#E10:1_1_1_28_2"><span class="lab"><font color="Green" title="E79">A909</font></span></a>, <a class="txt" href="#E16:1_1_1_28_2"><span class="lab"><font color="Green" title="E85">A915</font></span></a>, <a class="txt" href="#E1:1_1_1_28_2"><span class="lab"><font color="Green" title="E72">A902</font></span></a>, <a class="txt" href="#E5:1_1_1_28_2"><span class="lab"><font color="Green" title="E76">A906</font></span></a>, <a class="ref" href="jordan9.html#T23" title="JORDAN9:th.23">JORDAN9:23</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_2_1_1"/><span class="lab"><font color="Green" title="E86">A916</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">j</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_28_2_1_1"/><span class="lab"><font color="Green" title="E87">A917</font></span>: 
<font color="Maroon" title="c17">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a>, <a class="txt" href="#E8:1_1_1_28_2"><span class="lab"><font color="Green" title="E77">A907</font></span></a>, <a class="txt" href="#E13:1_1_1_28_2"><span class="lab"><font color="Green" title="E82">A912</font></span></a></span>;<br/><a NAME="E3:1_1_1_28_2_1_1"/>
<font color="Maroon" title="c17">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a>, <a class="txt" href="#E14:1_1_1_28_2"><span class="lab"><font color="Green" title="E83">A913</font></span></a></span>;<br/><a NAME="E4:1_1_1_28_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E88">A918</font></span>: 
<font color="Maroon" title="c17">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_28_2"><span class="lab"><font color="Green" title="E73">A903</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E2:1_1_1_28_2"><span class="lab"><font color="Green" title="E73">A903</font></span></a>, <a class="txt" href="#E14:1_1_1_28_2"><span class="lab"><font color="Green" title="E83">A913</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E2:1_1_1_28_2"><span class="lab"><font color="Green" title="E73">A903</font></span></a>, <a class="txt" href="#E4:1_1_1_28_2"><span class="lab"><font color="Green" title="E75">A905</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c19">j</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E16:1_1_1_28_2"><span class="lab"><font color="Green" title="E85">A915</font></span></a>, <a class="txt" href="#E12:1_1_1_28_2"><span class="lab"><font color="Green" title="E81">A911</font></span></a>, <a class="txt" href="#E1:1_1_1_28_2_1_1"><span class="lab"><font color="Green" title="E86">A916</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c19">j</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_28_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_1"><span class="lab"><font color="Green" title="E36">A647</font></span></a>, <a class="txt" href="#E16:1_1_1_28_2"><span class="lab"><font color="Green" title="E85">A915</font></span></a>, <a class="txt" href="#E12:1_1_1_28_2"><span class="lab"><font color="Green" title="E81">A911</font></span></a>, <a class="txt" href="#E4:1_1_1_28_2_1_1"><span class="lab"><font color="Green" title="E88">A918</font></span></a>, <a class="txt" href="#E2:1_1_1_28_2_1_1"><span class="lab"><font color="Green" title="E87">A917</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_2_1_2"/><span class="lab"><font color="Green" title="E86">A919</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_2_1_2_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_2_1_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_28_2_1_2_1_1"/>
( <font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font> or <font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font> )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_1_28_2"><span class="lab"><font color="Green" title="E80">A910</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_2_1_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E87">A920</font></span>: 
<font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:1_1_1_28_2_1_2_1_1_1"/><span class="lab"><font color="Green" title="E88">A921</font></span>: 
<font color="Maroon" title="c17">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a>, <a class="txt" href="#E13:1_1_1_28_2"><span class="lab"><font color="Green" title="E82">A912</font></span></a>, <a class="txt" href="#E15:1_1_1_28_2"><span class="lab"><font color="Green" title="E84">A914</font></span></a></span>;<br/><a NAME="E3:1_1_1_28_2_1_2_1_1_1"/>
<font color="Maroon" title="c17">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a>, <a class="txt" href="#E13:1_1_1_28_2"><span class="lab"><font color="Green" title="E82">A912</font></span></a>, <a class="txt" href="#E15:1_1_1_28_2"><span class="lab"><font color="Green" title="E84">A914</font></span></a></span>;<br/><a NAME="E4:1_1_1_28_2_1_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A922</font></span>: 
<font color="Maroon" title="c17">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_28_2"><span class="lab"><font color="Green" title="E73">A903</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E2:1_1_1_28_2"><span class="lab"><font color="Green" title="E73">A903</font></span></a>, <a class="txt" href="#E13:1_1_1_28_2"><span class="lab"><font color="Green" title="E82">A912</font></span></a>, <a class="txt" href="#E15:1_1_1_28_2"><span class="lab"><font color="Green" title="E84">A914</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E2:1_1_1_28_2"><span class="lab"><font color="Green" title="E73">A903</font></span></a>, <a class="txt" href="#E4:1_1_1_28_2"><span class="lab"><font color="Green" title="E75">A905</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">m</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:1_1_1"><span class="lab"><font color="Green" title="E39">A650</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E46:1_1_1"><span class="lab"><font color="Green" title="E40">A651</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E1:1_1_1_28_2_1_2"><span class="lab"><font color="Green" title="E86">A919</font></span></a>, <a class="txt" href="#E1:1_1_1_28_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E87">A920</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_28_2_1_2_1_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_1"><span class="lab"><font color="Green" title="E36">A647</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E46:1_1_1"><span class="lab"><font color="Green" title="E40">A651</font></span></a>, <a class="txt" href="#E50:1_1_1"><span class="lab"><font color="Green" title="E43">A654</font></span></a>, <a class="txt" href="#E4:1_1_1_28_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E89">A922</font></span></a>, <a class="txt" href="#E2:1_1_1_28_2_1_2_1_1_1"><span class="lab"><font color="Green" title="E88">A921</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_2_1_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E87">A923</font></span>: 
<font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c20">l</font> = <font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1;<br/><a NAME="E2:1_1_1_28_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E88">A924</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><a NAME="E3:1_1_1_28_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E89">A925</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E4:1_1_1_28_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E90">A926</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_28_2"><span class="lab"><font color="Green" title="E83">A913</font></span></a>, <a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E5:1_1_1_28_2_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A927</font></span>: 
<font color="Maroon" title="c17">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a>, <a class="txt" href="#E3:1_1_1_28_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E89">A925</font></span></a></span>;<br/><a NAME="E6:1_1_1_28_2_1_2_1_1_2"/><span class="lab"><font color="Green" title="E92">A928</font></span>: 
<font color="Maroon" title="c17">n</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_28_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E89">A925</font></span></a>, <a class="txt" href="#E4:1_1_1_28_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E90">A926</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E7:1_1_1_28_2_1_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c17">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/><a NAME="E8:1_1_1_28_2_1_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A929</font></span>: 
<font color="Maroon" title="c17">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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_28_2"><span class="lab"><font color="Green" title="E73">A903</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c17">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E2:1_1_1_28_2"><span class="lab"><font color="Green" title="E73">A903</font></span></a>, <a class="txt" href="#E6:1_1_1_28_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E92">A928</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c17">n</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E2:1_1_1_28_2"><span class="lab"><font color="Green" title="E73">A903</font></span></a>, <a class="txt" href="#E4:1_1_1_28_2"><span class="lab"><font color="Green" title="E75">A905</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E1:1_1_1_28_2_1_2"><span class="lab"><font color="Green" title="E86">A919</font></span></a>, <a class="txt" href="#E1:1_1_1_28_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E87">A923</font></span></a>, <a class="txt" href="#E2:1_1_1_28_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E88">A924</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c19">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E10:1_1_1_28_2_1_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_1"><span class="lab"><font color="Green" title="E36">A647</font></span></a>, <a class="txt" href="#E1:1_1_1_28_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E87">A923</font></span></a>, <a class="txt" href="#E8:1_1_1_28_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E93">A929</font></span></a>, <a class="txt" href="#E5:1_1_1_28_2_1_2_1_1_2"><span class="lab"><font color="Green" title="E91">A927</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E3:1_1_1_28_2_1_2"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E5:1_1_1_28"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_3"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:1_1_1_28_3"/>
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E70:1_1_1"><span class="lab"><font color="Green" title="E54">A703</font></span></a>, <a class="ref" href="xreal_1.html#T145" title="XREAL_1:th.145">XREAL_1:145</a></span>;<br/>
<a NAME="E2:1_1_1_28_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A930</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_28"><span class="lab"><font color="Green" title="E69">A899</font></span></a></span>;<br/>
<a NAME="E3:1_1_1_28_3"/><span class="lab"><font color="Green" title="E73">A931</font></span>: 
1 <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c10">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_28"><span class="lab"><font color="Green" title="E70">A900</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E4:1_1_1_28_3"/><span class="lab"><font color="Green" title="E74">A932</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:1_1_1"><span class="lab"><font color="Green" title="E40">A651</font></span></a>, <a class="txt" href="#E1:1_1_1_28"><span class="lab"><font color="Green" title="E69">A899</font></span></a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c17">l</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">a</font> = <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c19">rg</font> =  <a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c20">p</font> = <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1;<br/>
<span class="kw">set </span><font color="Maroon" title="c21">q</font> = <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2;<br/>
<a NAME="E5:1_1_1_28_3"/><span class="lab"><font color="Green" title="E75">A933</font></span>: 
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:1_1_1"><span class="lab"><font color="Green" title="E53">A702</font></span></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></span>;<br/>
<a NAME="E6:1_1_1_28_3"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:1_1_1"><span class="lab"><font color="Green" title="E53">A702</font></span></a>, <a class="ref" href="nat_d.html#T42" title="NAT_D:th.42">NAT_D:42</a></span>;<br/>
<a NAME="E7:1_1_1_28_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A934</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/>
<a NAME="E8:1_1_1_28_3"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E9:1_1_1_28_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E10:1_1_1_28_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A935</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/>
<a NAME="E11:1_1_1_28_3"/><span class="lab"><font color="Green" title="E78">A936</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:1_1_1"><span class="lab"><font color="Green" title="E53">A702</font></span></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>;<br/>
<a NAME="E12:1_1_1_28_3"/><span class="kw">then </span>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T42" title="NAT_D:th.42">NAT_D:42</a></span>;<br/>
<a NAME="E13:1_1_1_28_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A937</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/>
<a NAME="E14:1_1_1_28_3"/><span class="lab"><font color="Green" title="E80">A938</font></span>: 
 <a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E71:1_1_1"><span class="lab"><font color="Green" title="E55">A704</font></span></a>, <a class="ref" href="jordan9.html#T5" title="JORDAN9:th.5">JORDAN9:5</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c22">p1</font>, <font color="Maroon" title="c23">p2</font>, <font color="Maroon" title="c24">q1</font>, <font color="Maroon" title="c25">q2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E16:1_1_1_28_3"/><span class="lab"><font color="Green" title="E81">A939</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">p1</font>,<font color="Maroon" title="c23">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E17:1_1_1_28_3"/><span class="lab"><font color="Green" title="E82">A940</font></span>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c22">p1</font>,<font color="Maroon" title="c23">p2</font>)
 <span class="kw">and </span><br/><a NAME="E18:1_1_1_28_3"/><span class="lab"><font color="Green" title="E83">A941</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c24">q1</font>,<font color="Maroon" title="c25">q2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E19:1_1_1_28_3"/><span class="lab"><font color="Green" title="E84">A942</font></span>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c24">q1</font>,<font color="Maroon" title="c25">q2</font>)
 <span class="kw">and </span><br/><a NAME="E20:1_1_1_28_3"/><span class="lab"><font color="Green" title="E85">A943</font></span>: 
( ( <font color="Maroon" title="c22">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q1</font> &amp; <font color="Maroon" title="c23">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q2</font> ) or ( <font color="Maroon" title="c22">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q1</font> &amp; <font color="Maroon" title="c23">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q2</font> ) or ( <font color="Maroon" title="c22">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q2</font> ) or ( <font color="Maroon" title="c22">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q1</font> &amp; <font color="Maroon" title="c23">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_1_28_3"><span class="lab"><font color="Green" title="E78">A936</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/>
<a NAME="E21:1_1_1_28_3"/><span class="lab"><font color="Green" title="E86">A944</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E22:1_1_1_28_3"/><span class="lab"><font color="Green" title="E87">A945</font></span>: 
<font color="Maroon" title="c23">p2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E23:1_1_1_28_3"/><span class="lab"><font color="Green" title="E88">A946</font></span>: 
<font color="Maroon" title="c22">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E24:1_1_1_28_3"/><span class="lab"><font color="Green" title="E89">A947</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c23">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/>
<a NAME="E25:1_1_1_28_3"/><span class="lab"><font color="Green" title="E90">A948</font></span>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:1_1_1"><span class="lab"><font color="Green" title="E39">A650</font></span></a>, <a class="txt" href="#E67:1_1_1"><span class="lab"><font color="Green" title="E52">A701</font></span></a>, <a class="ref" href="finseq_5.html#T65" title="FINSEQ_5:th.65">FINSEQ_5:65</a></span>;<br/>
<a NAME="E26:1_1_1_28_3"/>
<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> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T44" title="NAT_D:th.44">NAT_D:44</a></span>;<br/>
<a NAME="E27:1_1_1_28_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A949</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_28_3"><span class="lab"><font color="Green" title="E76">A934</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E92">A950</font></span>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 = 
<font color="Maroon" title="c15">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_28_3"><span class="lab"><font color="Green" title="E72">A930</font></span></a>, <a class="ref" href="finseq_5.html#T66" title="FINSEQ_5:th.66">FINSEQ_5:66</a></span>
<br/>.= 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:1_1_1_28_3"><span class="lab"><font color="Green" title="E91">A949</font></span></a>, <a class="ref" href="finseq_5.html#T27" title="FINSEQ_5:th.27">FINSEQ_5:27</a></span>
;<br/>
<a NAME="E29:1_1_1_28_3"/>
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="c15">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_1_28_3"><span class="lab"><font color="Green" title="E78">A936</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E30:1_1_1_28_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E93">A951</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<span class="lab"><font color="Green" title="E94">A952</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:1_1_1"><span class="lab"><font color="Green" title="E40">A651</font></span></a>, <a class="ref" href="nat_d.html#T38" title="NAT_D:th.38">NAT_D:38</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_28_3"><span class="lab"><font color="Green" title="E76">A934</font></span></a>, <a class="ref" href="nat_d.html#T37" title="NAT_D:th.37">NAT_D:37</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:1_1_1_28_3"><span class="lab"><font color="Green" title="E75">A933</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p4">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1</span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E63:1_1_1"><span class="lab"><font color="Green" title="E51">A700</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3"><span class="lab"><font color="Green" title="E73">A931</font></span></a>, <a class="ref" href="xreal_0.html#D2" title="XREAL_0:def.2">XREAL_0:def 2</a></span>
<br/>.= 
<font color="Maroon" title="c9">k</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1

;<br/>
<a NAME="E32:1_1_1_28_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E95">A953</font></span>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E67:1_1_1"><span class="lab"><font color="Green" title="E52">A701</font></span></a>, <a class="ref" href="finseq_5.html#T65" title="FINSEQ_5:th.65">FINSEQ_5:65</a></span>;<br/>
<a NAME="E33:1_1_1_28_3"/><span class="lab"><font color="Green" title="E96">A954</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">m</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:1_1_1"><span class="lab"><font color="Green" title="E40">A651</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/>
<a NAME="E34:1_1_1_28_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E97">A955</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_28"><span class="lab"><font color="Green" title="E70">A900</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E35:1_1_1_28_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E98">A956</font></span>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a></span>;<br/>
<a NAME="E36:1_1_1_28_3"/>
<font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E37:1_1_1_28_3"/><span class="kw">then </span>
<font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>;<br/>
<a NAME="E38:1_1_1_28_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E99">A957</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/>
<a NAME="E39:1_1_1_28_3"/>
 not <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c15">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_28"><span class="lab"><font color="Green" title="E71">A901</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a></span>;<br/>
<a NAME="E40:1_1_1_28_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E100">A958</font></span>: 
 not <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></span>;<br/>
<a NAME="E41:1_1_1_28_3"/><span class="lab"><font color="Green" title="E101">A959</font></span>: 
<font color="Maroon" title="c9">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E31:1_1_1_28_3"><span class="lab"><font color="Green" title="E94">A952</font></span></a></span>;<br/>
<a NAME="E42:1_1_1_28_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E102">A960</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c9">k</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/>
<a NAME="E43:1_1_1_28_3"/>
<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> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E44:1_1_1_28_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E103">A961</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_28_3"><span class="lab"><font color="Green" title="E76">A934</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E45:1_1_1_28_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E104">A962</font></span>: 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E41:1_1_1_28_3"><span class="lab"><font color="Green" title="E101">A959</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_28_3_3"/>
( ( <font color="Maroon" title="c22">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q1</font> &amp; <font color="Maroon" title="c23">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q2</font> ) or ( <font color="Maroon" title="c22">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q1</font> &amp; <font color="Maroon" title="c23">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q2</font> ) or ( <font color="Maroon" title="c22">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q2</font> ) or ( <font color="Maroon" title="c22">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q1</font> &amp; <font color="Maroon" title="c23">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_1_1_28_3"><span class="lab"><font color="Green" title="E85">A943</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_1"/><span class="lab"><font color="Green" title="E105">A963</font></span>: 
( <font color="Maroon" title="c22">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q1</font> &amp; <font color="Maroon" title="c23">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c26">a1</font>, <font color="Maroon" title="c27">a2</font>, <font color="Maroon" title="c28">p91</font>, <font color="Maroon" title="c29">p92</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:1_1_1_28_3_3_1"/><span class="lab"><font color="Green" title="E106">A964</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">a1</font>,<font color="Maroon" title="c27">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_28_3_3_1"/><span class="lab"><font color="Green" title="E107">A965</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c26">a1</font>,<font color="Maroon" title="c27">a2</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_28_3_3_1"/><span class="lab"><font color="Green" title="E108">A966</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c28">p91</font>,<font color="Maroon" title="c29">p92</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:1_1_1_28_3_3_1"/><span class="lab"><font color="Green" title="E109">A967</font></span>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c28">p91</font>,<font color="Maroon" title="c29">p92</font>)
 <span class="kw">and </span><br/><a NAME="E7:1_1_1_28_3_3_1"/><span class="lab"><font color="Green" title="E110">A968</font></span>: 
( ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/><a NAME="E8:1_1_1_28_3_3_1"/><span class="lab"><font color="Green" title="E111">A969</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E106">A964</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><span class="kw">thus </span><a NAME="E9:1_1_1_28_3_3_1"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_3_3_1_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_1_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_28_3_3_1_1_1"/>
( ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E110">A968</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_1_1_1_1"/><span class="lab"><font color="Green" title="E112">A970</font></span>: 
( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_28_3_3_1_1_1_1"/><span class="lab"><font color="Green" title="E113">A971</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><span class="lab"><font color="Green" title="E114">A972</font></span>: <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E62:1_1_1"><span class="lab"><font color="Green" title="E50">A699</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E2:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E113">A971</font></span></a></span>
;<br/><a NAME="E4:1_1_1_28_3_3_1_1_1_1"/><span class="lab"><font color="Green" title="E115">A973</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:1_1_1"><span class="lab"><font color="Green" title="E53">A702</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:1_1_1_28_3_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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_28_3"><span class="lab"><font color="Green" title="E72">A930</font></span></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A974</font></span>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c15">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E115">A973</font></span></a>, <a class="ref" href="finseq_5.html#T66" title="FINSEQ_5:th.66">FINSEQ_5:66</a></span>
<br/>.= 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_28_3"><span class="lab"><font color="Green" title="E74">A932</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E115">A973</font></span></a>, <a class="ref" href="finseq_5.html#T27" title="FINSEQ_5:th.27">FINSEQ_5:27</a></span>
;<br/><a NAME="E7:1_1_1_28_3_3_1_1_1_1"/><span class="lab"><font color="Green" title="E117">A975</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E88:1_1_1"><span class="lab"><font color="Green" title="E67">A897</font></span></a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></span>;<br/><span class="lab"><font color="Green" title="E118">A976</font></span>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c15">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:1_1_1"><span class="lab"><font color="Green" title="E39">A650</font></span></a>, <a class="txt" href="#E77:1_1_1"><span class="lab"><font color="Green" title="E60">A737</font></span></a>, <a class="txt" href="#E67:1_1_1"><span class="lab"><font color="Green" title="E52">A701</font></span></a>, <a class="ref" href="finseq_5.html#T65" title="FINSEQ_5:th.65">FINSEQ_5:65</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T65" title="FINSEQ_5:th.65">FINSEQ_5:65</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>
;<br/><a NAME="E9:1_1_1_28_3_3_1_1_1_1"/><span class="lab"><font color="Green" title="E119">A977</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">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="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_1"><span class="lab"><font color="Green" title="E31">A564</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E57:1_1_1"><span class="lab"><font color="Green" title="E47">A658</font></span></a></span>;<br/><a NAME="E10:1_1_1_28_3_3_1_1_1_1"/><span class="lab"><font color="Green" title="E120">A978</font></span>: 
<font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E108">A966</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E109">A967</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E112">A970</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E11:1_1_1_28_3_3_1_1_1_1"/><span class="lab"><font color="Green" title="E121">A979</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E46:1_1_1"><span class="lab"><font color="Green" title="E40">A651</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a></span>;<br/><a NAME="E12:1_1_1_28_3_3_1_1_1_1"/><span class="lab"><font color="Green" title="E122">A980</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c30">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span>;<br/><a NAME="E13:1_1_1_28_3_3_1_1_1_1"/><span class="lab"><font color="Green" title="E123">A981</font></span>: 
<font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c27">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E14:1_1_1_28_3_3_1_1_1_1"/><span class="lab"><font color="Green" title="E124">A982</font></span>: 
<font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E108">A966</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E109">A967</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E112">A970</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E15:1_1_1_28_3_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A983</font></span>: 
<font color="Maroon" title="c23">p2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/> <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c22">p1</font>,<font color="Maroon" title="c23">p2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E31:1_1_1_28_3"><span class="lab"><font color="Green" title="E94">A952</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E28:1_1_1_28_3"><span class="lab"><font color="Green" title="E92">A950</font></span></a>, <a class="txt" href="#E32:1_1_1_28_3"><span class="lab"><font color="Green" title="E95">A953</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E18:1_1_1_28_3"><span class="lab"><font color="Green" title="E83">A941</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3"><span class="lab"><font color="Green" title="E84">A942</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E105">A963</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>
<br/>.= 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E106">A964</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E107">A965</font></span></a>, <a class="txt" href="#E10:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E120">A978</font></span></a>, <a class="txt" href="#E14:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E124">A982</font></span></a>, <a class="txt" href="#E12:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E122">A980</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E114">A972</font></span></a>, <a class="txt" href="#E11:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E121">A979</font></span></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></span>
;<br/><a NAME="E17:1_1_1_28_3_3_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:1_1_1"><span class="lab"><font color="Green" title="E27">A513</font></span></a>, <a class="txt" href="#E3:1_1_1_28"><span class="lab"><font color="Green" title="E70">A900</font></span></a>, <a class="txt" href="#E45:1_1_1_28_3"><span class="lab"><font color="Green" title="E104">A962</font></span></a></span>;<br/><a NAME="E18:1_1_1_28_3_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A984</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3"><span class="lab"><font color="Green" title="E74">A932</font></span></a>, <a class="txt" href="#E9:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E119">A977</font></span></a>, <a class="ref" href="gobrd13.html#T43" title="GOBRD13:th.43">GOBRD13:43</a></span>;<br/><a NAME="E19:1_1_1_28_3_3_1_1_1_1"/><span class="lab"><font color="Green" title="E127">A985</font></span>: 
<font color="Maroon" title="c23">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E124">A982</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E20:1_1_1_28_3_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A986</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c23">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E106">A964</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E107">A965</font></span></a>, <a class="txt" href="#E13:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E123">A981</font></span></a>, <a class="txt" href="#E18:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E126">A984</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>;<br/><a NAME="E21:1_1_1_28_3_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A987</font></span>: 
<font color="Maroon" title="c22">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E22:1_1_1_28_3_3_1_1_1_1"/>
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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="c22">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c23">p2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3"><span class="lab"><font color="Green" title="E74">A932</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E106">A964</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E107">A965</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E127">A985</font></span></a>, <a class="txt" href="#E13:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E123">A981</font></span></a>, <a class="txt" href="#E18:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E126">A984</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>;<br/><a NAME="E23:1_1_1_28_3_3_1_1_1_1"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c22">p1</font>,<font color="Maroon" title="c27">a2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_28_3"><span class="lab"><font color="Green" title="E80">A938</font></span></a>, <a class="txt" href="#E13:1_1_1_28_3"><span class="lab"><font color="Green" title="E79">A937</font></span></a>, <a class="txt" href="#E30:1_1_1_28_3"><span class="lab"><font color="Green" title="E93">A951</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E20:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E128">A986</font></span></a>, <a class="txt" href="#E15:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E125">A983</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E116">A974</font></span></a>, <a class="txt" href="#E8:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E118">A976</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>;<br/><a NAME="E24:1_1_1_28_3_3_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_28_3"><span class="lab"><font color="Green" title="E86">A944</font></span></a>, <a class="txt" href="#E22:1_1_1_28_3"><span class="lab"><font color="Green" title="E87">A945</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E107">A965</font></span></a>, <a class="txt" href="#E8:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E111">A969</font></span></a>, <a class="txt" href="#E10:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E120">A978</font></span></a>, <a class="txt" href="#E14:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E124">A982</font></span></a>, <a class="txt" href="#E21:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E129">A987</font></span></a>, <a class="ref" href="jordan9.html#T20" title="JORDAN9:th.20">JORDAN9:20</a></span>;<br/><a NAME="E25:1_1_1_28_3_3_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A988</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_1_28_3"><span class="lab"><font color="Green" title="E100">A958</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E26:1_1_1_28_3_3_1_1_1_1"/><span class="lab"><font color="Green" title="E131">A989</font></span>: 
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a></span>;<br/><a NAME="E27:1_1_1_28_3_3_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_28_3"><span class="lab"><font color="Green" title="E80">A938</font></span></a>, <a class="txt" href="#E13:1_1_1_28_3"><span class="lab"><font color="Green" title="E79">A937</font></span></a>, <a class="txt" href="#E30:1_1_1_28_3"><span class="lab"><font color="Green" title="E93">A951</font></span></a>, <a class="ref" href="jordan9.html#T27" title="JORDAN9:th.27">JORDAN9:27</a></span>;<br/><span class="kw">hence </span><a NAME="E28:1_1_1_28_3_3_1_1_1_1"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E58:1_1_1"><span class="lab"><font color="Green" title="E48">A659</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E40:1_1_1_28_3"><span class="lab"><font color="Green" title="E100">A958</font></span></a>, <a class="txt" href="#E25:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E130">A988</font></span></a>, <a class="txt" href="#E7:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E117">A975</font></span></a>, <a class="txt" href="#E26:1_1_1_28_3_3_1_1_1_1"><span class="lab"><font color="Green" title="E131">A989</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_1_1_1_2"/><span class="lab"><font color="Green" title="E112">A990</font></span>: 
( <font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_28_3_3_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E108">A966</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E109">A967</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E3:1_1_1_28_3_3_1_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A991</font></span>: 
<font color="Maroon" title="c24">q1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E105">A963</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E4:1_1_1_28_3_3_1_1_1_2"/>
<font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E108">A966</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E109">A967</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_1_1_1_2"><span class="lab"><font color="Green" title="E112">A990</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">then </span> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c26">a1</font>,<font color="Maroon" title="c27">a2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E31:1_1_1_28_3"><span class="lab"><font color="Green" title="E94">A952</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E28:1_1_1_28_3"><span class="lab"><font color="Green" title="E92">A950</font></span></a>, <a class="txt" href="#E32:1_1_1_28_3"><span class="lab"><font color="Green" title="E95">A953</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E18:1_1_1_28_3"><span class="lab"><font color="Green" title="E83">A941</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3"><span class="lab"><font color="Green" title="E84">A942</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E105">A963</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_1_1_1_2"><span class="lab"><font color="Green" title="E113">A991</font></span></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E106">A964</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E107">A965</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E108">A966</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E109">A967</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_1_1_1_2"><span class="lab"><font color="Green" title="E112">A990</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_28_3_3_1_1_1_2"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E41:1_1_1_28_3"><span class="lab"><font color="Green" title="E101">A959</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E35:1_1_1_28_3"><span class="lab"><font color="Green" title="E98">A956</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_1_1_1_3"/><span class="lab"><font color="Green" title="E112">A992</font></span>: 
( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_28_3_3_1_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A993</font></span>: 
<font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E108">A966</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E109">A967</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E3:1_1_1_28_3_3_1_1_1_3"/>
<font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E108">A966</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E109">A967</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_1_1_1_3"><span class="lab"><font color="Green" title="E112">A992</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">then </span> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c22">p1</font>,<font color="Maroon" title="c23">p2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:1_1_1"><span class="lab"><font color="Green" title="E39">A650</font></span></a>, <a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E67:1_1_1"><span class="lab"><font color="Green" title="E52">A701</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E106">A964</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E107">A965</font></span></a>, <a class="txt" href="#E2:1_1_1_28_3_3_1_1_1_3"><span class="lab"><font color="Green" title="E113">A993</font></span></a>, <a class="ref" href="finseq_5.html#T65" title="FINSEQ_5:th.65">FINSEQ_5:65</a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E31:1_1_1_28_3"><span class="lab"><font color="Green" title="E94">A952</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E28:1_1_1_28_3"><span class="lab"><font color="Green" title="E92">A950</font></span></a>, <a class="txt" href="#E32:1_1_1_28_3"><span class="lab"><font color="Green" title="E95">A953</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E18:1_1_1_28_3"><span class="lab"><font color="Green" title="E83">A941</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3"><span class="lab"><font color="Green" title="E84">A942</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E105">A963</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:1_1_1_28_3_3_1_1_1_3"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E45:1_1_1_28_3"><span class="lab"><font color="Green" title="E104">A962</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_1_1_1_4"/><span class="lab"><font color="Green" title="E112">A994</font></span>: 
( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_28_3_3_1_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A995</font></span>: 
<font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E105">A963</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E108">A966</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E109">A967</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E3:1_1_1_28_3_3_1_1_1_4"/><span class="lab"><font color="Green" title="E114">A996</font></span>: 
<font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E105">A963</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E108">A966</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E109">A967</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_1_1_1_4"><span class="lab"><font color="Green" title="E112">A994</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E62:1_1_1"><span class="lab"><font color="Green" title="E50">A699</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3"><span class="lab"><font color="Green" title="E84">A942</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_1"><span class="lab"><font color="Green" title="E107">A965</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_1_1_1_4"><span class="lab"><font color="Green" title="E114">A996</font></span></a>, <a class="txt" href="#E2:1_1_1_28_3_3_1_1_1_4"><span class="lab"><font color="Green" title="E113">A995</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E42:1_1_1_28_3"><span class="lab"><font color="Green" title="E102">A960</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E28:1_1_1_28_3"><span class="lab"><font color="Green" title="E92">A950</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E5:1_1_1_28_3_3_1_1_1_4"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_1"><span class="lab"><font color="Green" title="E36">A647</font></span></a>, <a class="txt" href="#E42:1_1_1_28_3"><span class="lab"><font color="Green" title="E102">A960</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E38:1_1_1_28_3"><span class="lab"><font color="Green" title="E99">A957</font></span></a>, <a class="txt" href="#E10:1_1_1_28_3"><span class="lab"><font color="Green" title="E77">A935</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_2"/><span class="lab"><font color="Green" title="E105">A997</font></span>: 
( <font color="Maroon" title="c22">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q1</font> &amp; <font color="Maroon" title="c23">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c26">a1</font>, <font color="Maroon" title="c27">a2</font>, <font color="Maroon" title="c28">p91</font>, <font color="Maroon" title="c29">p92</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:1_1_1_28_3_3_2"/><span class="lab"><font color="Green" title="E106">A998</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">a1</font>,<font color="Maroon" title="c27">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_28_3_3_2"/><span class="lab"><font color="Green" title="E107">A999</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c26">a1</font>,<font color="Maroon" title="c27">a2</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_28_3_3_2"/><span class="lab"><font color="Green" title="E108">A1000</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c28">p91</font>,<font color="Maroon" title="c29">p92</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:1_1_1_28_3_3_2"/><span class="lab"><font color="Green" title="E109">A1001</font></span>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c28">p91</font>,<font color="Maroon" title="c29">p92</font>)
 <span class="kw">and </span><br/><a NAME="E7:1_1_1_28_3_3_2"/><span class="lab"><font color="Green" title="E110">A1002</font></span>: 
( ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/><a NAME="E8:1_1_1_28_3_3_2"/><span class="lab"><font color="Green" title="E111">A1003</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E106">A998</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E9:1_1_1_28_3_3_2"/><span class="lab"><font color="Green" title="E112">A1004</font></span>: 
<font color="Maroon" title="c27">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E106">A998</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E10:1_1_1_28_3_3_2"/><span class="lab"><font color="Green" title="E113">A1005</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c26">a1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E106">A998</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><span class="kw">thus </span><a NAME="E11:1_1_1_28_3_3_2"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_3_3_2_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_2_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_28_3_3_2_1_1"/>
( ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E110">A1002</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_2_1_1_1"/><span class="lab"><font color="Green" title="E114">A1006</font></span>: 
( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_28_3_3_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E108">A1000</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E109">A1001</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E3:1_1_1_28_3_3_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A1007</font></span>: 
<font color="Maroon" title="c25">q2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E105">A997</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E4:1_1_1_28_3_3_2_1_1_1"/><span class="lab"><font color="Green" title="E116">A1008</font></span>: 
<font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E108">A1000</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E109">A1001</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_2_1_1_1"><span class="lab"><font color="Green" title="E114">A1006</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c26">a1</font>,<font color="Maroon" title="c27">a2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E106">A998</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E107">A999</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E108">A1000</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E109">A1001</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_2_1_1_1"><span class="lab"><font color="Green" title="E114">A1006</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E31:1_1_1_28_3"><span class="lab"><font color="Green" title="E94">A952</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E28:1_1_1_28_3"><span class="lab"><font color="Green" title="E92">A950</font></span></a>, <a class="txt" href="#E32:1_1_1_28_3"><span class="lab"><font color="Green" title="E95">A953</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E18:1_1_1_28_3"><span class="lab"><font color="Green" title="E83">A941</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3"><span class="lab"><font color="Green" title="E84">A942</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E105">A997</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_2_1_1_1"><span class="lab"><font color="Green" title="E116">A1008</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_2_1_1_1"><span class="lab"><font color="Green" title="E115">A1007</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_28_3_3_2_1_1_1"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E45:1_1_1_28_3"><span class="lab"><font color="Green" title="E104">A962</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_2_1_1_2"/><span class="lab"><font color="Green" title="E114">A1009</font></span>: 
( <font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_28_3_3_2_1_1_2"/><span class="lab"><font color="Green" title="E115">A1010</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><span class="lab"><font color="Green" title="E116">A1011</font></span>: <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E62:1_1_1"><span class="lab"><font color="Green" title="E50">A699</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E2:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E115">A1010</font></span></a></span>
;<br/><a NAME="E4:1_1_1_28_3_3_2_1_1_2"/><span class="lab"><font color="Green" title="E117">A1012</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:1_1_1"><span class="lab"><font color="Green" title="E53">A702</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:1_1_1_28_3_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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_28_3"><span class="lab"><font color="Green" title="E72">A930</font></span></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E118">A1013</font></span>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c15">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E117">A1012</font></span></a>, <a class="ref" href="finseq_5.html#T66" title="FINSEQ_5:th.66">FINSEQ_5:66</a></span>
<br/>.= 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_28_3"><span class="lab"><font color="Green" title="E74">A932</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E117">A1012</font></span></a>, <a class="ref" href="finseq_5.html#T27" title="FINSEQ_5:th.27">FINSEQ_5:27</a></span>
;<br/><a NAME="E7:1_1_1_28_3_3_2_1_1_2"/><span class="lab"><font color="Green" title="E119">A1014</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E88:1_1_1"><span class="lab"><font color="Green" title="E67">A897</font></span></a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></span>;<br/><a NAME="E8:1_1_1_28_3_3_2_1_1_2"/><span class="lab"><font color="Green" title="E120">A1015</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">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="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_1"><span class="lab"><font color="Green" title="E31">A564</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E57:1_1_1"><span class="lab"><font color="Green" title="E47">A658</font></span></a></span>;<br/><a NAME="E9:1_1_1_28_3_3_2_1_1_2"/><span class="lab"><font color="Green" title="E121">A1016</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a></span>;<br/><a NAME="E10:1_1_1_28_3_3_2_1_1_2"/><span class="lab"><font color="Green" title="E122">A1017</font></span>: 
<font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E108">A1000</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E109">A1001</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E114">A1009</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="lab"><font color="Green" title="E123">A1018</font></span>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c15">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:1_1_1"><span class="lab"><font color="Green" title="E39">A650</font></span></a>, <a class="txt" href="#E77:1_1_1"><span class="lab"><font color="Green" title="E60">A737</font></span></a>, <a class="txt" href="#E67:1_1_1"><span class="lab"><font color="Green" title="E52">A701</font></span></a>, <a class="ref" href="finseq_5.html#T65" title="FINSEQ_5:th.65">FINSEQ_5:65</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T65" title="FINSEQ_5:th.65">FINSEQ_5:65</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>
;<br/><span class="kw">set </span><font color="Maroon" title="c30">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span>;<br/><a NAME="E12:1_1_1_28_3_3_2_1_1_2"/><span class="lab"><font color="Green" title="E124">A1019</font></span>: 
<font color="Maroon" title="c22">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c22">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E13:1_1_1_28_3_3_2_1_1_2"/><span class="lab"><font color="Green" title="E125">A1020</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E46:1_1_1"><span class="lab"><font color="Green" title="E40">A651</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a></span>;<br/><a NAME="E14:1_1_1_28_3_3_2_1_1_2"/><span class="lab"><font color="Green" title="E126">A1021</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c27">a2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E111">A1003</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E15:1_1_1_28_3_3_2_1_1_2"/><span class="lab"><font color="Green" title="E127">A1022</font></span>: 
<font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E108">A1000</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E109">A1001</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E114">A1009</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E16:1_1_1_28_3_3_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A1023</font></span>: 
<font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/> <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c22">p1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">p2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E31:1_1_1_28_3"><span class="lab"><font color="Green" title="E94">A952</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E28:1_1_1_28_3"><span class="lab"><font color="Green" title="E92">A950</font></span></a>, <a class="txt" href="#E32:1_1_1_28_3"><span class="lab"><font color="Green" title="E95">A953</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E18:1_1_1_28_3"><span class="lab"><font color="Green" title="E83">A941</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3"><span class="lab"><font color="Green" title="E84">A942</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E105">A997</font></span></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>
<br/>.= 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E106">A998</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E107">A999</font></span></a>, <a class="txt" href="#E15:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E127">A1022</font></span></a>, <a class="txt" href="#E10:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E122">A1017</font></span></a>, <a class="txt" href="#E9:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E121">A1016</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E116">A1011</font></span></a>, <a class="txt" href="#E13:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E125">A1020</font></span></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></span>
;<br/><a NAME="E18:1_1_1_28_3_3_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:1_1_1"><span class="lab"><font color="Green" title="E27">A513</font></span></a>, <a class="txt" href="#E3:1_1_1_28"><span class="lab"><font color="Green" title="E70">A900</font></span></a>, <a class="txt" href="#E45:1_1_1_28_3"><span class="lab"><font color="Green" title="E104">A962</font></span></a></span>;<br/><a NAME="E19:1_1_1_28_3_3_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A1024</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3"><span class="lab"><font color="Green" title="E74">A932</font></span></a>, <a class="txt" href="#E8:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E120">A1015</font></span></a>, <a class="ref" href="gobrd13.html#T43" title="GOBRD13:th.43">GOBRD13:43</a></span>;<br/><a NAME="E20:1_1_1_28_3_3_2_1_1_2"/><span class="lab"><font color="Green" title="E130">A1025</font></span>: 
<font color="Maroon" title="c26">a1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E21:1_1_1_28_3_3_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A1026</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">p1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">p2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E106">A998</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E107">A999</font></span></a>, <a class="txt" href="#E15:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E127">A1022</font></span></a>, <a class="txt" href="#E12:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E124">A1019</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E129">A1024</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>;<br/><a NAME="E22:1_1_1_28_3_3_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E132">A1027</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c27">a2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E122">A1017</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E23:1_1_1_28_3_3_2_1_1_2"/>
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">p1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">p2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3"><span class="lab"><font color="Green" title="E74">A932</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E106">A998</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E107">A999</font></span></a>, <a class="txt" href="#E15:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E127">A1022</font></span></a>, <a class="txt" href="#E20:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E130">A1025</font></span></a>, <a class="txt" href="#E12:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E124">A1019</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E129">A1024</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>;<br/><a NAME="E24:1_1_1_28_3_3_2_1_1_2"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c26">a1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c27">a2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_28_3"><span class="lab"><font color="Green" title="E80">A938</font></span></a>, <a class="txt" href="#E13:1_1_1_28_3"><span class="lab"><font color="Green" title="E79">A937</font></span></a>, <a class="txt" href="#E30:1_1_1_28_3"><span class="lab"><font color="Green" title="E93">A951</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E10:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E122">A1017</font></span></a>, <a class="txt" href="#E21:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E131">A1026</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E128">A1023</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E118">A1013</font></span></a>, <a class="txt" href="#E14:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E126">A1021</font></span></a>, <a class="txt" href="#E11:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E123">A1018</font></span></a>, <a class="ref" href="gobrd13.html#T21" title="GOBRD13:th.21">GOBRD13:21</a></span>;<br/><a NAME="E25:1_1_1_28_3_3_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1_28_3"><span class="lab"><font color="Green" title="E88">A946</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E107">A999</font></span></a>, <a class="txt" href="#E10:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E113">A1005</font></span></a>, <a class="txt" href="#E9:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E112">A1004</font></span></a>, <a class="txt" href="#E15:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E127">A1022</font></span></a>, <a class="txt" href="#E14:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E126">A1021</font></span></a>, <a class="txt" href="#E22:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E132">A1027</font></span></a>, <a class="ref" href="jordan9.html#T20" title="JORDAN9:th.20">JORDAN9:20</a></span>;<br/><a NAME="E26:1_1_1_28_3_3_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E133">A1028</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_1_28_3"><span class="lab"><font color="Green" title="E100">A958</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E27:1_1_1_28_3_3_2_1_1_2"/><span class="lab"><font color="Green" title="E134">A1029</font></span>: 
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a></span>;<br/><a NAME="E28:1_1_1_28_3_3_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_28_3"><span class="lab"><font color="Green" title="E80">A938</font></span></a>, <a class="txt" href="#E13:1_1_1_28_3"><span class="lab"><font color="Green" title="E79">A937</font></span></a>, <a class="txt" href="#E30:1_1_1_28_3"><span class="lab"><font color="Green" title="E93">A951</font></span></a>, <a class="ref" href="jordan9.html#T27" title="JORDAN9:th.27">JORDAN9:27</a></span>;<br/><span class="kw">hence </span><a NAME="E29:1_1_1_28_3_3_2_1_1_2"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E58:1_1_1"><span class="lab"><font color="Green" title="E48">A659</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E40:1_1_1_28_3"><span class="lab"><font color="Green" title="E100">A958</font></span></a>, <a class="txt" href="#E26:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E133">A1028</font></span></a>, <a class="txt" href="#E7:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E119">A1014</font></span></a>, <a class="txt" href="#E27:1_1_1_28_3_3_2_1_1_2"><span class="lab"><font color="Green" title="E134">A1029</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_2_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_2_1_1_3"/><span class="lab"><font color="Green" title="E114">A1030</font></span>: 
( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_28_3_3_2_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A1031</font></span>: 
<font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E105">A997</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E108">A1000</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E109">A1001</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E3:1_1_1_28_3_3_2_1_1_3"/><span class="lab"><font color="Green" title="E116">A1032</font></span>: 
<font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E105">A997</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E108">A1000</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E109">A1001</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_2_1_1_3"><span class="lab"><font color="Green" title="E114">A1030</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E62:1_1_1"><span class="lab"><font color="Green" title="E50">A699</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3"><span class="lab"><font color="Green" title="E84">A942</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E107">A999</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_2_1_1_3"><span class="lab"><font color="Green" title="E116">A1032</font></span></a>, <a class="txt" href="#E2:1_1_1_28_3_3_2_1_1_3"><span class="lab"><font color="Green" title="E115">A1031</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E42:1_1_1_28_3"><span class="lab"><font color="Green" title="E102">A960</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E28:1_1_1_28_3"><span class="lab"><font color="Green" title="E92">A950</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E5:1_1_1_28_3_3_2_1_1_3"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_1"><span class="lab"><font color="Green" title="E36">A647</font></span></a>, <a class="txt" href="#E42:1_1_1_28_3"><span class="lab"><font color="Green" title="E102">A960</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E38:1_1_1_28_3"><span class="lab"><font color="Green" title="E99">A957</font></span></a>, <a class="txt" href="#E10:1_1_1_28_3"><span class="lab"><font color="Green" title="E77">A935</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_2_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_2_1_1_4"/><span class="lab"><font color="Green" title="E114">A1033</font></span>: 
( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_28_3_3_2_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E115">A1034</font></span>: 
<font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E108">A1000</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E109">A1001</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E3:1_1_1_28_3_3_2_1_1_4"/><span class="lab"><font color="Green" title="E116">A1035</font></span>: 
<font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E108">A1000</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E109">A1001</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_2_1_1_4"><span class="lab"><font color="Green" title="E114">A1033</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c22">p1</font>,<font color="Maroon" title="c23">p2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E31:1_1_1_28_3"><span class="lab"><font color="Green" title="E94">A952</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E28:1_1_1_28_3"><span class="lab"><font color="Green" title="E92">A950</font></span></a>, <a class="txt" href="#E32:1_1_1_28_3"><span class="lab"><font color="Green" title="E95">A953</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E18:1_1_1_28_3"><span class="lab"><font color="Green" title="E83">A941</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3"><span class="lab"><font color="Green" title="E84">A942</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E105">A997</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:1_1_1"><span class="lab"><font color="Green" title="E39">A650</font></span></a>, <a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E67:1_1_1"><span class="lab"><font color="Green" title="E52">A701</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E106">A998</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_2"><span class="lab"><font color="Green" title="E107">A999</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_2_1_1_4"><span class="lab"><font color="Green" title="E116">A1035</font></span></a>, <a class="txt" href="#E2:1_1_1_28_3_3_2_1_1_4"><span class="lab"><font color="Green" title="E115">A1034</font></span></a>, <a class="ref" href="finseq_5.html#T65" title="FINSEQ_5:th.65">FINSEQ_5:65</a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>
;<br/><span class="kw">hence </span><a NAME="E5:1_1_1_28_3_3_2_1_1_4"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E41:1_1_1_28_3"><span class="lab"><font color="Green" title="E101">A959</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E35:1_1_1_28_3"><span class="lab"><font color="Green" title="E98">A956</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_3"/><span class="lab"><font color="Green" title="E105">A1036</font></span>: 
( <font color="Maroon" title="c22">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c23">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q2</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c26">a1</font>, <font color="Maroon" title="c27">a2</font>, <font color="Maroon" title="c28">p91</font>, <font color="Maroon" title="c29">p92</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:1_1_1_28_3_3_3"/><span class="lab"><font color="Green" title="E106">A1037</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">a1</font>,<font color="Maroon" title="c27">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_28_3_3_3"/><span class="lab"><font color="Green" title="E107">A1038</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c26">a1</font>,<font color="Maroon" title="c27">a2</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_28_3_3_3"/><span class="lab"><font color="Green" title="E108">A1039</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c28">p91</font>,<font color="Maroon" title="c29">p92</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:1_1_1_28_3_3_3"/><span class="lab"><font color="Green" title="E109">A1040</font></span>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c28">p91</font>,<font color="Maroon" title="c29">p92</font>)
 <span class="kw">and </span><br/><a NAME="E7:1_1_1_28_3_3_3"/><span class="lab"><font color="Green" title="E110">A1041</font></span>: 
( ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/><a NAME="E8:1_1_1_28_3_3_3"/><span class="lab"><font color="Green" title="E111">A1042</font></span>: 
<font color="Maroon" title="c26">a1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E106">A1037</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><span class="kw">thus </span><a NAME="E9:1_1_1_28_3_3_3"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_3_3_3_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_3_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_28_3_3_3_1_1"/>
( ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E110">A1041</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_3_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_3_1_1_1"/><span class="lab"><font color="Green" title="E112">A1043</font></span>: 
( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_28_3_3_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E108">A1039</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E109">A1040</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E3:1_1_1_28_3_3_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A1044</font></span>: 
<font color="Maroon" title="c25">q2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c27">a2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E105">A1036</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E4:1_1_1_28_3_3_3_1_1_1"/>
<font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E108">A1039</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E109">A1040</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_3_1_1_1"><span class="lab"><font color="Green" title="E112">A1043</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E5:1_1_1_28_3_3_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A1045</font></span>: 
<font color="Maroon" title="c24">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E105">A1036</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c24">q1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c25">q2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E31:1_1_1_28_3"><span class="lab"><font color="Green" title="E94">A952</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E28:1_1_1_28_3"><span class="lab"><font color="Green" title="E92">A950</font></span></a>, <a class="txt" href="#E32:1_1_1_28_3"><span class="lab"><font color="Green" title="E95">A953</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E18:1_1_1_28_3"><span class="lab"><font color="Green" title="E83">A941</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3"><span class="lab"><font color="Green" title="E84">A942</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E105">A1036</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E106">A1037</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E107">A1038</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E108">A1039</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E109">A1040</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_3_1_1_1"><span class="lab"><font color="Green" title="E112">A1043</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_3_1_1_1"><span class="lab"><font color="Green" title="E114">A1045</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_3_1_1_1"><span class="lab"><font color="Green" title="E113">A1044</font></span></a>, <a class="ref" href="gobrd13.html#T21" title="GOBRD13:th.21">GOBRD13:21</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:1_1_1_28_3_3_3_1_1_1"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E41:1_1_1_28_3"><span class="lab"><font color="Green" title="E101">A959</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E35:1_1_1_28_3"><span class="lab"><font color="Green" title="E98">A956</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_3_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_3_1_1_2"/><span class="lab"><font color="Green" title="E112">A1046</font></span>: 
( <font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_28_3_3_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A1047</font></span>: 
<font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E108">A1039</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E109">A1040</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E3:1_1_1_28_3_3_3_1_1_2"/><span class="lab"><font color="Green" title="E114">A1048</font></span>: 
<font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E108">A1039</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E109">A1040</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_3_1_1_2"><span class="lab"><font color="Green" title="E112">A1046</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E62:1_1_1"><span class="lab"><font color="Green" title="E50">A699</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3"><span class="lab"><font color="Green" title="E84">A942</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E105">A1036</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E107">A1038</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_3_1_1_2"><span class="lab"><font color="Green" title="E114">A1048</font></span></a>, <a class="txt" href="#E2:1_1_1_28_3_3_3_1_1_2"><span class="lab"><font color="Green" title="E113">A1047</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E42:1_1_1_28_3"><span class="lab"><font color="Green" title="E102">A960</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E28:1_1_1_28_3"><span class="lab"><font color="Green" title="E92">A950</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E5:1_1_1_28_3_3_3_1_1_2"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_1"><span class="lab"><font color="Green" title="E36">A647</font></span></a>, <a class="txt" href="#E42:1_1_1_28_3"><span class="lab"><font color="Green" title="E102">A960</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E38:1_1_1_28_3"><span class="lab"><font color="Green" title="E99">A957</font></span></a>, <a class="txt" href="#E10:1_1_1_28_3"><span class="lab"><font color="Green" title="E77">A935</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_3_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_3_1_1_3"/><span class="lab"><font color="Green" title="E112">A1049</font></span>: 
( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_28_3_3_3_1_1_3"/><span class="lab"><font color="Green" title="E113">A1050</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><span class="lab"><font color="Green" title="E114">A1051</font></span>: <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E62:1_1_1"><span class="lab"><font color="Green" title="E50">A699</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E2:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E113">A1050</font></span></a></span>
;<br/><a NAME="E4:1_1_1_28_3_3_3_1_1_3"/><span class="lab"><font color="Green" title="E115">A1052</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:1_1_1"><span class="lab"><font color="Green" title="E53">A702</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:1_1_1_28_3_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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_28_3"><span class="lab"><font color="Green" title="E72">A930</font></span></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A1053</font></span>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c15">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E115">A1052</font></span></a>, <a class="ref" href="finseq_5.html#T66" title="FINSEQ_5:th.66">FINSEQ_5:66</a></span>
<br/>.= 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_28_3"><span class="lab"><font color="Green" title="E74">A932</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E115">A1052</font></span></a>, <a class="ref" href="finseq_5.html#T27" title="FINSEQ_5:th.27">FINSEQ_5:27</a></span>
;<br/><a NAME="E7:1_1_1_28_3_3_3_1_1_3"/><span class="lab"><font color="Green" title="E117">A1054</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E88:1_1_1"><span class="lab"><font color="Green" title="E67">A897</font></span></a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c30">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span>;<br/><a NAME="E8:1_1_1_28_3_3_3_1_1_3"/><span class="lab"><font color="Green" title="E118">A1055</font></span>: 
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a></span>;<br/><a NAME="E9:1_1_1_28_3_3_3_1_1_3"/><span class="lab"><font color="Green" title="E119">A1056</font></span>: 
<font color="Maroon" title="c22">p1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E105">A1036</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E10:1_1_1_28_3_3_3_1_1_3"/><span class="lab"><font color="Green" title="E120">A1057</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">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="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_1"><span class="lab"><font color="Green" title="E31">A564</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E57:1_1_1"><span class="lab"><font color="Green" title="E47">A658</font></span></a></span>;<br/><a NAME="E11:1_1_1_28_3_3_3_1_1_3"/><span class="lab"><font color="Green" title="E121">A1058</font></span>: 
<font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E108">A1039</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E109">A1040</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E112">A1049</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E12:1_1_1_28_3_3_3_1_1_3"/><span class="lab"><font color="Green" title="E122">A1059</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E46:1_1_1"><span class="lab"><font color="Green" title="E40">A651</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a></span>;<br/><a NAME="E13:1_1_1_28_3_3_3_1_1_3"/><span class="lab"><font color="Green" title="E123">A1060</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a></span>;<br/><a NAME="E14:1_1_1_28_3_3_3_1_1_3"/><span class="lab"><font color="Green" title="E124">A1061</font></span>: 
<font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E108">A1039</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E109">A1040</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E112">A1049</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/> <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c24">q1</font>,<font color="Maroon" title="c25">q2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E31:1_1_1_28_3"><span class="lab"><font color="Green" title="E94">A952</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E28:1_1_1_28_3"><span class="lab"><font color="Green" title="E92">A950</font></span></a>, <a class="txt" href="#E32:1_1_1_28_3"><span class="lab"><font color="Green" title="E95">A953</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E18:1_1_1_28_3"><span class="lab"><font color="Green" title="E83">A941</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3"><span class="lab"><font color="Green" title="E84">A942</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E105">A1036</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>
<br/>.= 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E105">A1036</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E106">A1037</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E107">A1038</font></span></a>, <a class="txt" href="#E11:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E121">A1058</font></span></a>, <a class="txt" href="#E14:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E124">A1061</font></span></a>, <a class="txt" href="#E9:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E119">A1056</font></span></a>, <a class="txt" href="#E13:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E123">A1060</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E114">A1051</font></span></a>, <a class="txt" href="#E12:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E122">A1059</font></span></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></span>
;<br/><a NAME="E16:1_1_1_28_3_3_3_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:1_1_1"><span class="lab"><font color="Green" title="E27">A513</font></span></a>, <a class="txt" href="#E3:1_1_1_28"><span class="lab"><font color="Green" title="E70">A900</font></span></a>, <a class="txt" href="#E45:1_1_1_28_3"><span class="lab"><font color="Green" title="E104">A962</font></span></a></span>;<br/><a NAME="E17:1_1_1_28_3_3_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E125">A1062</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3"><span class="lab"><font color="Green" title="E74">A932</font></span></a>, <a class="txt" href="#E10:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E120">A1057</font></span></a>, <a class="ref" href="gobrd13.html#T43" title="GOBRD13:th.43">GOBRD13:43</a></span>;<br/><a NAME="E18:1_1_1_28_3_3_3_1_1_3"/>
<font color="Maroon" title="c22">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c22">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T29" title="XREAL_1:th.29">XREAL_1:29</a></span>;<br/><a NAME="E19:1_1_1_28_3_3_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E126">A1063</font></span>: 
<font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c22">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E121">A1058</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E20:1_1_1_28_3_3_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A1064</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c22">p1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c23">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E106">A1037</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E107">A1038</font></span></a>, <a class="txt" href="#E14:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E124">A1061</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E125">A1062</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>;<br/><a NAME="E21:1_1_1_28_3_3_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E128">A1065</font></span>: 
<font color="Maroon" title="c23">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E22:1_1_1_28_3_3_3_1_1_3"/>
<font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E124">A1061</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E23:1_1_1_28_3_3_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A1066</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font 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">p1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c23">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3"><span class="lab"><font color="Green" title="E74">A932</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E106">A1037</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E107">A1038</font></span></a>, <a class="txt" href="#E14:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E124">A1061</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E126">A1063</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E125">A1062</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>;<br/><span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c15">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:1_1_1"><span class="lab"><font color="Green" title="E39">A650</font></span></a>, <a class="txt" href="#E77:1_1_1"><span class="lab"><font color="Green" title="E60">A737</font></span></a>, <a class="txt" href="#E67:1_1_1"><span class="lab"><font color="Green" title="E52">A701</font></span></a>, <a class="ref" href="finseq_5.html#T65" title="FINSEQ_5:th.65">FINSEQ_5:65</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T65" title="FINSEQ_5:th.65">FINSEQ_5:65</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>
;<br/><a NAME="E25:1_1_1_28_3_3_3_1_1_3"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c22">p1</font>,<font color="Maroon" title="c23">p2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_28_3"><span class="lab"><font color="Green" title="E80">A938</font></span></a>, <a class="txt" href="#E13:1_1_1_28_3"><span class="lab"><font color="Green" title="E79">A937</font></span></a>, <a class="txt" href="#E30:1_1_1_28_3"><span class="lab"><font color="Green" title="E93">A951</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E23:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E129">A1066</font></span></a>, <a class="txt" href="#E20:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E127">A1064</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E116">A1053</font></span></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></span>;<br/><a NAME="E26:1_1_1_28_3_3_3_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_28_3"><span class="lab"><font color="Green" title="E86">A944</font></span></a>, <a class="txt" href="#E24:1_1_1_28_3"><span class="lab"><font color="Green" title="E89">A947</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E107">A1038</font></span></a>, <a class="txt" href="#E8:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E111">A1042</font></span></a>, <a class="txt" href="#E11:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E121">A1058</font></span></a>, <a class="txt" href="#E14:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E124">A1061</font></span></a>, <a class="txt" href="#E21:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E128">A1065</font></span></a>, <a class="ref" href="jordan9.html#T20" title="JORDAN9:th.20">JORDAN9:20</a></span>;<br/><a NAME="E27:1_1_1_28_3_3_3_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A1067</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_1_28_3"><span class="lab"><font color="Green" title="E100">A958</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E28:1_1_1_28_3_3_3_1_1_3"/>
<span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_28_3"><span class="lab"><font color="Green" title="E80">A938</font></span></a>, <a class="txt" href="#E13:1_1_1_28_3"><span class="lab"><font color="Green" title="E79">A937</font></span></a>, <a class="txt" href="#E30:1_1_1_28_3"><span class="lab"><font color="Green" title="E93">A951</font></span></a>, <a class="ref" href="jordan9.html#T27" title="JORDAN9:th.27">JORDAN9:27</a></span>;<br/><span class="kw">hence </span><a NAME="E29:1_1_1_28_3_3_3_1_1_3"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E58:1_1_1"><span class="lab"><font color="Green" title="E48">A659</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E40:1_1_1_28_3"><span class="lab"><font color="Green" title="E100">A958</font></span></a>, <a class="txt" href="#E27:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E130">A1067</font></span></a>, <a class="txt" href="#E7:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E117">A1054</font></span></a>, <a class="txt" href="#E8:1_1_1_28_3_3_3_1_1_3"><span class="lab"><font color="Green" title="E118">A1055</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_3_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_3_1_1_4"/><span class="lab"><font color="Green" title="E112">A1068</font></span>: 
( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_28_3_3_3_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E108">A1039</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E109">A1040</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E3:1_1_1_28_3_3_3_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A1069</font></span>: 
<font color="Maroon" title="c24">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c26">a1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E105">A1036</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E4:1_1_1_28_3_3_3_1_1_4"/>
<font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E108">A1039</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E109">A1040</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_3_1_1_4"><span class="lab"><font color="Green" title="E112">A1068</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="kw">then </span> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c24">q1</font>,<font color="Maroon" title="c25">q2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:1_1_1"><span class="lab"><font color="Green" title="E39">A650</font></span></a>, <a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E67:1_1_1"><span class="lab"><font color="Green" title="E52">A701</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E105">A1036</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E106">A1037</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E107">A1038</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E108">A1039</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E109">A1040</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_3_1_1_4"><span class="lab"><font color="Green" title="E112">A1068</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_3_1_1_4"><span class="lab"><font color="Green" title="E113">A1069</font></span></a>, <a class="ref" href="finseq_5.html#T65" title="FINSEQ_5:th.65">FINSEQ_5:65</a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E31:1_1_1_28_3"><span class="lab"><font color="Green" title="E94">A952</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E28:1_1_1_28_3"><span class="lab"><font color="Green" title="E92">A950</font></span></a>, <a class="txt" href="#E32:1_1_1_28_3"><span class="lab"><font color="Green" title="E95">A953</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E18:1_1_1_28_3"><span class="lab"><font color="Green" title="E83">A941</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3"><span class="lab"><font color="Green" title="E84">A942</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_3"><span class="lab"><font color="Green" title="E105">A1036</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_28_3_3_3_1_1_4"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E45:1_1_1_28_3"><span class="lab"><font color="Green" title="E104">A962</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_4"/><span class="lab"><font color="Green" title="E105">A1070</font></span>: 
( <font color="Maroon" title="c22">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q1</font> &amp; <font color="Maroon" title="c23">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c26">a1</font>, <font color="Maroon" title="c27">a2</font>, <font color="Maroon" title="c28">p91</font>, <font color="Maroon" title="c29">p92</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E3:1_1_1_28_3_3_4"/><span class="lab"><font color="Green" title="E106">A1071</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c26">a1</font>,<font color="Maroon" title="c27">a2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:1_1_1_28_3_3_4"/><span class="lab"><font color="Green" title="E107">A1072</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c26">a1</font>,<font color="Maroon" title="c27">a2</font>)
 <span class="kw">and </span><br/><a NAME="E5:1_1_1_28_3_3_4"/><span class="lab"><font color="Green" title="E108">A1073</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c28">p91</font>,<font color="Maroon" title="c29">p92</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E6:1_1_1_28_3_3_4"/><span class="lab"><font color="Green" title="E109">A1074</font></span>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<font color="Maroon" title="c28">p91</font>,<font color="Maroon" title="c29">p92</font>)
 <span class="kw">and </span><br/><a NAME="E7:1_1_1_28_3_3_4"/><span class="lab"><font color="Green" title="E110">A1075</font></span>: 
( ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="ref" href="jordan8.html#T3" title="JORDAN8:th.3">JORDAN8:3</a></span>;<br/><a NAME="E8:1_1_1_28_3_3_4"/><span class="lab"><font color="Green" title="E111">A1076</font></span>: 
<font color="Maroon" title="c27">a2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E106">A1071</font></span></a>, <a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><span class="kw">thus </span><a NAME="E9:1_1_1_28_3_3_4"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_3_3_4_1"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_4_1_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:1_1_1_28_3_3_4_1_1"/>
( ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> ) or ( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E110">A1075</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_4_1_1_1"/><span class="lab"><font color="Green" title="E112">A1077</font></span>: 
( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_28_3_3_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A1078</font></span>: 
<font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E108">A1073</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E109">A1074</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E3:1_1_1_28_3_3_4_1_1_1"/><span class="lab"><font color="Green" title="E114">A1079</font></span>: 
<font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E108">A1073</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E109">A1074</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_4_1_1_1"><span class="lab"><font color="Green" title="E112">A1077</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E62:1_1_1"><span class="lab"><font color="Green" title="E50">A699</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3"><span class="lab"><font color="Green" title="E84">A942</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E105">A1070</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E107">A1072</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_4_1_1_1"><span class="lab"><font color="Green" title="E114">A1079</font></span></a>, <a class="txt" href="#E2:1_1_1_28_3_3_4_1_1_1"><span class="lab"><font color="Green" title="E113">A1078</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E42:1_1_1_28_3"><span class="lab"><font color="Green" title="E102">A960</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E28:1_1_1_28_3"><span class="lab"><font color="Green" title="E92">A950</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">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="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a></span>
;<br/><span class="kw">hence </span><a NAME="E5:1_1_1_28_3_3_4_1_1_1"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_1"><span class="lab"><font color="Green" title="E36">A647</font></span></a>, <a class="txt" href="#E42:1_1_1_28_3"><span class="lab"><font color="Green" title="E102">A960</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E38:1_1_1_28_3"><span class="lab"><font color="Green" title="E99">A957</font></span></a>, <a class="txt" href="#E10:1_1_1_28_3"><span class="lab"><font color="Green" title="E77">A935</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_4_1_1_2"/><span class="lab"><font color="Green" title="E112">A1080</font></span>: 
( <font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_28_3_3_4_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E108">A1073</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E109">A1074</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E3:1_1_1_28_3_3_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A1081</font></span>: 
<font color="Maroon" title="c27">a2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E105">A1070</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E4:1_1_1_28_3_3_4_1_1_2"/>
<font color="Maroon" title="c26">a1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E108">A1073</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E109">A1074</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_4_1_1_2"><span class="lab"><font color="Green" title="E112">A1080</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E5:1_1_1_28_3_3_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E114">A1082</font></span>: 
<font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">q1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E105">A1070</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c26">a1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c27">a2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E106">A1071</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E107">A1072</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E108">A1073</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E109">A1074</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_4_1_1_2"><span class="lab"><font color="Green" title="E112">A1080</font></span></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E31:1_1_1_28_3"><span class="lab"><font color="Green" title="E94">A952</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E28:1_1_1_28_3"><span class="lab"><font color="Green" title="E92">A950</font></span></a>, <a class="txt" href="#E32:1_1_1_28_3"><span class="lab"><font color="Green" title="E95">A953</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E18:1_1_1_28_3"><span class="lab"><font color="Green" title="E83">A941</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3"><span class="lab"><font color="Green" title="E84">A942</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E105">A1070</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_4_1_1_2"><span class="lab"><font color="Green" title="E114">A1082</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_4_1_1_2"><span class="lab"><font color="Green" title="E113">A1081</font></span></a>, <a class="ref" href="gobrd13.html#T21" title="GOBRD13:th.21">GOBRD13:21</a></span>
;<br/><span class="kw">hence </span><a NAME="E7:1_1_1_28_3_3_4_1_1_2"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E45:1_1_1_28_3"><span class="lab"><font color="Green" title="E104">A962</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_4_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_4_1_1_3"/><span class="lab"><font color="Green" title="E112">A1083</font></span>: 
( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_28_3_3_4_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E108">A1073</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E109">A1074</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E3:1_1_1_28_3_3_4_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A1084</font></span>: 
<font color="Maroon" title="c27">a2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E105">A1070</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><a NAME="E4:1_1_1_28_3_3_4_1_1_3"/><span class="lab"><font color="Green" title="E114">A1085</font></span>: 
<font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E108">A1073</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E109">A1074</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_4_1_1_3"><span class="lab"><font color="Green" title="E112">A1083</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/> <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<font color="Maroon" title="c24">q1</font>,<font color="Maroon" title="c25">q2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E31:1_1_1_28_3"><span class="lab"><font color="Green" title="E94">A952</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E28:1_1_1_28_3"><span class="lab"><font color="Green" title="E92">A950</font></span></a>, <a class="txt" href="#E32:1_1_1_28_3"><span class="lab"><font color="Green" title="E95">A953</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E18:1_1_1_28_3"><span class="lab"><font color="Green" title="E83">A941</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3"><span class="lab"><font color="Green" title="E84">A942</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E105">A1070</font></span></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></span>
<br/>.= 
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:1_1_1"><span class="lab"><font color="Green" title="E39">A650</font></span></a>, <a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E67:1_1_1"><span class="lab"><font color="Green" title="E52">A701</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E105">A1070</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E106">A1071</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E107">A1072</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E108">A1073</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E109">A1074</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_4_1_1_3"><span class="lab"><font color="Green" title="E112">A1083</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_4_1_1_3"><span class="lab"><font color="Green" title="E114">A1085</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_4_1_1_3"><span class="lab"><font color="Green" title="E113">A1084</font></span></a>, <a class="ref" href="finseq_5.html#T65" title="FINSEQ_5:th.65">FINSEQ_5:65</a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></span>
;<br/><span class="kw">hence </span><a NAME="E6:1_1_1_28_3_3_4_1_1_3"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E41:1_1_1_28_3"><span class="lab"><font color="Green" title="E101">A959</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E35:1_1_1_28_3"><span class="lab"><font color="Green" title="E98">A956</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_4_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:1_1_1_28_3_3_4_1_1_4"/><span class="lab"><font color="Green" title="E112">A1086</font></span>: 
( <font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">p91</font> &amp; <font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c29">p92</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span><br/><div class="add"><a NAME="E2:1_1_1_28_3_3_4_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E113">A1087</font></span>: 
<font color="Maroon" title="c27">a2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E108">A1073</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E109">A1074</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/><a NAME="E3:1_1_1_28_3_3_4_1_1_4"/><span class="lab"><font color="Green" title="E114">A1088</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E46:1_1_1"><span class="lab"><font color="Green" title="E40">A651</font></span></a>, <a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a></span>;<br/><a NAME="E4:1_1_1_28_3_3_4_1_1_4"/><span class="lab"><font color="Green" title="E115">A1089</font></span>: 
2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c15">g</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E69:1_1_1"><span class="lab"><font color="Green" title="E53">A702</font></span></a>, <a class="ref" href="finseq_3.html#T25" title="FINSEQ_3:th.25">FINSEQ_3:25</a></span>;<br/><a NAME="E5:1_1_1_28_3_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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_28_3"><span class="lab"><font color="Green" title="E72">A930</font></span></a>, <a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E116">A1090</font></span>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<font color="Maroon" title="c15">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E115">A1089</font></span></a>, <a class="ref" href="finseq_5.html#T66" title="FINSEQ_5:th.66">FINSEQ_5:66</a></span>
<br/>.= 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1_28_3"><span class="lab"><font color="Green" title="E74">A932</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E115">A1089</font></span></a>, <a class="ref" href="finseq_5.html#T27" title="FINSEQ_5:th.27">FINSEQ_5:27</a></span>
;<br/><a NAME="E7:1_1_1_28_3_3_4_1_1_4"/><span class="lab"><font color="Green" title="E117">A1091</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">p1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_28_3"><span class="lab"><font color="Green" title="E86">A944</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a></span>;<br/><a NAME="E8:1_1_1_28_3_3_4_1_1_4"/><span class="lab"><font color="Green" title="E118">A1092</font></span>: 
<font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/><span class="lab"><font color="Green" title="E119">A1093</font></span>: <font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E62:1_1_1"><span class="lab"><font color="Green" title="E50">A699</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E8:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E118">A1092</font></span></a></span>
;<br/><a NAME="E10:1_1_1_28_3_3_4_1_1_4"/><span class="lab"><font color="Green" title="E120">A1094</font></span>: 
<font color="Maroon" title="c23">p2</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c25">q2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E105">A1070</font></span></a>, <a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c30">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span>;<br/><a NAME="E11:1_1_1_28_3_3_4_1_1_4"/><span class="lab"><font color="Green" title="E121">A1095</font></span>: 
<font color="Maroon" title="c23">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><span class="lab"><font color="Green" title="E122">A1096</font></span>: <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 = 
<font color="Maroon" title="c15">g</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E45:1_1_1"><span class="lab"><font color="Green" title="E39">A650</font></span></a>, <a class="txt" href="#E77:1_1_1"><span class="lab"><font color="Green" title="E60">A737</font></span></a>, <a class="txt" href="#E67:1_1_1"><span class="lab"><font color="Green" title="E52">A701</font></span></a>, <a class="ref" href="finseq_5.html#T65" title="FINSEQ_5:th.65">FINSEQ_5:65</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T65" title="FINSEQ_5:th.65">FINSEQ_5:65</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#D3" title="FINSEQ_5:def.3">FINSEQ_5:def 3</a></span>
;<br/><a NAME="E13:1_1_1_28_3_3_4_1_1_4"/><span class="lab"><font color="Green" title="E123">A1097</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:1_1_1"><span class="lab"><font color="Green" title="E17">A176</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a></span>;<br/><a NAME="E14:1_1_1_28_3_3_4_1_1_4"/><span class="lab"><font color="Green" title="E124">A1098</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c9">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="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_1"><span class="lab"><font color="Green" title="E31">A564</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E57:1_1_1"><span class="lab"><font color="Green" title="E47">A658</font></span></a></span>;<br/><a NAME="E15:1_1_1_28_3_3_4_1_1_4"/><span class="lab"><font color="Green" title="E125">A1099</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c12">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E88:1_1_1"><span class="lab"><font color="Green" title="E67">A897</font></span></a>, <a class="ref" href="sppol_2.html#T22" title="SPPOL_2:th.22">SPPOL_2:22</a></span>;<br/><a NAME="E16:1_1_1_28_3_3_4_1_1_4"/><span class="lab"><font color="Green" title="E126">A1100</font></span>: 
<font color="Maroon" title="c26">a1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c22">p1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E5:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E108">A1073</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E109">A1074</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E112">A1086</font></span></a>, <a class="ref" href="goboard1.html#T5" title="GOBOARD1:th.5">GOBOARD1:5</a></span>;<br/> <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<font color="Maroon" title="c12">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) = 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c24">q1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c25">q2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E31:1_1_1_28_3"><span class="lab"><font color="Green" title="E94">A952</font></span></a>, <a class="txt" href="#E44:1_1_1_28_3"><span class="lab"><font color="Green" title="E103">A961</font></span></a>, <a class="txt" href="#E28:1_1_1_28_3"><span class="lab"><font color="Green" title="E92">A950</font></span></a>, <a class="txt" href="#E32:1_1_1_28_3"><span class="lab"><font color="Green" title="E95">A953</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E18:1_1_1_28_3"><span class="lab"><font color="Green" title="E83">A941</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3"><span class="lab"><font color="Green" title="E84">A942</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E105">A1070</font></span></a>, <a class="ref" href="gobrd13.html#T21" title="GOBRD13:th.21">GOBRD13:21</a></span>
<br/>.= 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <font color="Maroon" title="c10">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E1:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E105">A1070</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E106">A1071</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E107">A1072</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E126">A1100</font></span></a>, <a class="txt" href="#E2:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E113">A1087</font></span></a>, <a class="txt" href="#E10:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E120">A1094</font></span></a>, <a class="txt" href="#E13:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E123">A1097</font></span></a>, <a class="txt" href="#E9:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E119">A1093</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E114">A1088</font></span></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></span>
;<br/><a NAME="E18:1_1_1_28_3_3_4_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:1_1_1"><span class="lab"><font color="Green" title="E27">A513</font></span></a>, <a class="txt" href="#E3:1_1_1_28"><span class="lab"><font color="Green" title="E70">A900</font></span></a>, <a class="txt" href="#E45:1_1_1_28_3"><span class="lab"><font color="Green" title="E104">A962</font></span></a></span>;<br/><a NAME="E19:1_1_1_28_3_3_4_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E127">A1101</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3"><span class="lab"><font color="Green" title="E74">A932</font></span></a>, <a class="txt" href="#E14:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E124">A1098</font></span></a>, <a class="ref" href="gobrd13.html#T43" title="GOBRD13:th.43">GOBRD13:43</a></span>;<br/><a NAME="E20:1_1_1_28_3_3_4_1_1_4"/><span class="lab"><font color="Green" title="E128">A1102</font></span>: 
<font color="Maroon" title="c27">a2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c23">p2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E113">A1087</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E21:1_1_1_28_3_3_4_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E129">A1103</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c22">p1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c23">p2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E106">A1071</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E107">A1072</font></span></a>, <a class="txt" href="#E11:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E121">A1095</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E127">A1101</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>;<br/><a NAME="E22:1_1_1_28_3_3_4_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E130">A1104</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c22">p1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="matrix_1.html#T38" title="MATRIX_1:th.38">MATRIX_1:38</a></span>;<br/><a NAME="E23:1_1_1_28_3_3_4_1_1_4"/>
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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="c22">p1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c23">p2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E25:1_1_1_28_3"><span class="lab"><font color="Green" title="E90">A948</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3"><span class="lab"><font color="Green" title="E74">A932</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E3:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E106">A1071</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E107">A1072</font></span></a>, <a class="txt" href="#E20:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E128">A1102</font></span></a>, <a class="txt" href="#E11:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E121">A1095</font></span></a>, <a class="txt" href="#E19:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E127">A1101</font></span></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></span>;<br/><a NAME="E24:1_1_1_28_3_3_4_1_1_4"/><span class="kw">then </span>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> (<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c22">p1</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<font color="Maroon" title="c23">p2</font>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_28_3"><span class="lab"><font color="Green" title="E80">A938</font></span></a>, <a class="txt" href="#E13:1_1_1_28_3"><span class="lab"><font color="Green" title="E79">A937</font></span></a>, <a class="txt" href="#E30:1_1_1_28_3"><span class="lab"><font color="Green" title="E93">A951</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3"><span class="lab"><font color="Green" title="E81">A939</font></span></a>, <a class="txt" href="#E17:1_1_1_28_3"><span class="lab"><font color="Green" title="E82">A940</font></span></a>, <a class="txt" href="#E21:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E129">A1103</font></span></a>, <a class="txt" href="#E6:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E116">A1090</font></span></a>, <a class="txt" href="#E12:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E122">A1096</font></span></a>, <a class="txt" href="#E7:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E117">A1091</font></span></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></span>;<br/><a NAME="E25:1_1_1_28_3_3_4_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1_28_3"><span class="lab"><font color="Green" title="E88">A946</font></span></a>, <a class="txt" href="#E24:1_1_1_28_3"><span class="lab"><font color="Green" title="E89">A947</font></span></a>, <a class="txt" href="#E4:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E107">A1072</font></span></a>, <a class="txt" href="#E8:1_1_1_28_3_3_4"><span class="lab"><font color="Green" title="E111">A1076</font></span></a>, <a class="txt" href="#E16:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E126">A1100</font></span></a>, <a class="txt" href="#E2:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E113">A1087</font></span></a>, <a class="txt" href="#E22:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E130">A1104</font></span></a>, <a class="txt" href="#E7:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E117">A1091</font></span></a>, <a class="ref" href="jordan9.html#T20" title="JORDAN9:th.20">JORDAN9:20</a></span>;<br/><a NAME="E26:1_1_1_28_3_3_4_1_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E131">A1105</font></span>: 
<font color="Maroon" title="c12">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">m</font> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E40:1_1_1_28_3"><span class="lab"><font color="Green" title="E100">A958</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><a NAME="E27:1_1_1_28_3_3_4_1_1_4"/><span class="lab"><font color="Green" title="E132">A1106</font></span>: 
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a></span>;<br/><a NAME="E28:1_1_1_28_3_3_4_1_1_4"/>
<span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> (<span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span></span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:1_1_1_28_3"><span class="lab"><font color="Green" title="E80">A938</font></span></a>, <a class="txt" href="#E13:1_1_1_28_3"><span class="lab"><font color="Green" title="E79">A937</font></span></a>, <a class="txt" href="#E30:1_1_1_28_3"><span class="lab"><font color="Green" title="E93">A951</font></span></a>, <a class="ref" href="jordan9.html#T27" title="JORDAN9:th.27">JORDAN9:27</a></span>;<br/><span class="kw">hence </span><a NAME="E29:1_1_1_28_3_3_4_1_1_4"/>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1" title="GOBOARD9:func.1">Rev</a> <font color="Maroon" title="c15">g</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E49:1_1_1"><span class="lab"><font color="Green" title="E42">A653</font></span></a>, <a class="txt" href="#E58:1_1_1"><span class="lab"><font color="Green" title="E48">A659</font></span></a>, <a class="txt" href="#E33:1_1_1_28_3"><span class="lab"><font color="Green" title="E96">A954</font></span></a>, <a class="txt" href="#E34:1_1_1_28_3"><span class="lab"><font color="Green" title="E97">A955</font></span></a>, <a class="txt" href="#E40:1_1_1_28_3"><span class="lab"><font color="Green" title="E100">A958</font></span></a>, <a class="txt" href="#E26:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E131">A1105</font></span></a>, <a class="txt" href="#E15:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E125">A1099</font></span></a>, <a class="txt" href="#E27:1_1_1_28_3_3_4_1_1_4"><span class="lab"><font color="Green" title="E132">A1106</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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

</div><span class="kw">end;</span></div>
<a NAME="E6:1_1_1_28"/><span class="kw">then </span>
<font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c15">g</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#T23" title="GOBOARD9:th.23">GOBOARD9:23</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:1_1_1_28"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:1_1_1"><span class="lab"><font color="Green" title="E63">A875</font></span></a>, <a class="txt" href="#E84:1_1_1"><span class="lab"><font color="Green" title="E64">A884</font></span></a>, <a class="txt" href="#E89:1_1_1"><span class="lab"><font color="Green" title="E68">A898</font></span></a>, <a class="txt" href="#E87:1_1_1"><span class="lab"><font color="Green" title="E66">A886</font></span></a>, <a class="ref" href="jordan9.html#T1" title="JORDAN9:th.1">JORDAN9:1</a>, <a class="ref" href="sprect_4.html#T6" title="SPRECT_4:th.6">SPRECT_4:6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="lab"><font color="Green" title="E69">A1107</font></span>: <font color="Maroon" title="c15">g</font> = 
<font color="Maroon" title="c12">f</font> <a href="rfinseq.html#K2" title="RFINSEQ:func.2">/^</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T232" title="XREAL_1:th.232">XREAL_1:232</a></span>
<br/>.= 
<font color="Maroon" title="c12">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_5.html#T28" title="FINSEQ_5:th.28">FINSEQ_5:28</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">f</font> = <font color="Maroon" title="c12">f</font> as   non  <a href="funct_1.html#V3" title="FUNCT_1:attr.3">constant</a>   <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a>  <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a> ;<br/>
<a NAME="E93:1_1_1"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E70">A1108</font></span>: <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</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="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 1</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T16" title="FINSEQ_4:th.16">FINSEQ_4:16</a></span>
<br/>.= 
<font color="Maroon" title="c17">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E39:1_1_1"><span class="lab"><font color="Green" title="E35">A646</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a></span>
;<br/>
<a NAME="E95:1_1_1"/>
<font color="Maroon" title="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</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"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span></span><a href="finseq_4.html#K2" title="FINSEQ_4:func.2">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1"><span class="lab"><font color="Green" title="E10">A156</font></span></a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E71">A1109</font></span>: <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</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"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</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="c7">F</font> <a href="gobrd13.html#K1" title="GOBRD13:func.1">.</a> 2</span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_4.html#T17" title="FINSEQ_4:th.17">FINSEQ_4:17</a></span>
<br/>.= 
<font color="Maroon" title="c17">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E53:1_1_1"><span class="lab"><font color="Green" title="E45">A656</font></span></a>, <a class="txt" href="#E34:1_1_1"><span class="lab"><font color="Green" title="E32">A565</font></span></a></span>
;<br/>
<a NAME="E97:1_1_1"/><span class="lab"><font color="Green" title="E72">A1110</font></span>: 
2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T49" title="JORDAN1H:th.49">JORDAN1H:49</a></span>;<br/>
<a NAME="E98:1_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="1_1_1_32"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:1_1_1_32"/>
 <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c17">f</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="goboard9.html#D1" title="GOBOARD9:def.1">GOBOARD9:def 1</a></span>;<br/>
<a NAME="E2:1_1_1_32"/><span class="kw">then </span>
<font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c17">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E83:1_1_1"><span class="lab"><font color="Green" title="E63">A875</font></span></a>, <a class="txt" href="#E87:1_1_1"><span class="lab"><font color="Green" title="E66">A886</font></span></a>, <a class="txt" href="#E91:1_1_1"><span class="lab"><font color="Green" title="E69">A1107</font></span></a>, <a class="ref" href="goboard9.html#T4" title="GOBOARD9:th.4">GOBOARD9:4</a></span>;<br/>
<a NAME="E3:1_1_1_32"/><span class="kw">then </span>
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c17">f</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="gobrd14.html#T14" title="GOBRD14:th.14">GOBRD14:14</a>, <a class="ref" href="xboole_1.html#T63" title="XBOOLE_1:th.63">XBOOLE_1:63</a></span>;<br/>
<a NAME="E4:1_1_1_32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A1111</font></span>: 
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c17">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T23" title="SUBSET_1:th.23">SUBSET_1:23</a></span>;<br/>
<a NAME="E5:1_1_1_32"/><span class="lab"><font color="Green" title="E74">A1112</font></span>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">f</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T65" title="JORDAN2C:th.65">JORDAN2C:65</a></span>;<br/>
<a NAME="E6:1_1_1_32"/>
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">f</font></span>)</span> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T65" title="JORDAN2C:th.65">JORDAN2C:65</a>, <a class="ref" href="jordan2c.html#T68" title="JORDAN2C:th.68">JORDAN2C:68</a></span>;<br/>
<a NAME="E7:1_1_1_32"/><span class="kw">then </span>
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">f</font></span>)</span> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">f</font></span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#D3" title="JORDAN2C:def.3">JORDAN2C:def 3</a></span>;<br/>
<a NAME="E8:1_1_1_32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E75">A1113</font></span>: 
(  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">f</font></span>)</span> <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="c17">f</font> or  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">L~</a> <font color="Maroon" title="c17">f</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <font color="Maroon" title="c17">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan1h.html#T24" title="JORDAN1H:th.24">JORDAN1H:24</a></span>;<br/>
<a NAME="E9:1_1_1_32"/><span class="lab"><font color="Green" title="E76">A1114</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E97:1_1_1"><span class="lab"><font color="Green" title="E72">A1110</font></span></a>, <a class="ref" href="xreal_1.html#T235" title="XREAL_1:th.235">XREAL_1:235</a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c18">W</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <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> : <font color="Olive" title="b1">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">C</font> </span> } </span> ;<br/>
<a NAME="E10:1_1_1_32"/><span class="lab"><font color="Green" title="E77">A1115</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c17">f</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="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c17">f</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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T16" title="TOPS_1:th.16">TOPS_1:16</a></span>;<br/>
<a NAME="E11:1_1_1_32"/><span class="lab"><font color="Green" title="E78">A1116</font></span>: 
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <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> : <font color="Olive" title="b1">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">C</font> </span> } </span> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#D4" title="JORDAN2C:def.4">JORDAN2C:def 4</a></span>;<br/>
<a NAME="E12:1_1_1_32"/><span class="lab"><font color="Green" title="E79">A1117</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c17">f</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="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E52:1_1_1"><span class="lab"><font color="Green" title="E44">A655</font></span></a>, <a class="ref" href="jordan9.html#T9" title="JORDAN9:th.9">JORDAN9:9</a></span>;<br/>
<a NAME="E13:1_1_1_32"/><span class="lab"><font color="Green" title="E80">A1118</font></span>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="ref" href="jordan11.html#T9" title="JORDAN11:th.9">JORDAN11:9</a></span>;<br/>
<a NAME="E14:1_1_1_32"/>
 <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"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</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="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:1"><span class="lab"><font color="Green" title="E1">A1</font></span></a>, <a class="ref" href="jordan11.html#T6" title="JORDAN11:th.6">JORDAN11:6</a></span>;<br/>
<a NAME="E15:1_1_1_32"/><span class="kw">then </span>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c17">f</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="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:1_1_1"><span class="lab"><font color="Green" title="E5">A5</font></span></a>, <a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a>, <a class="txt" href="#E52:1_1_1"><span class="lab"><font color="Green" title="E44">A655</font></span></a>, <a class="txt" href="#E94:1_1_1"><span class="lab"><font color="Green" title="E70">A1108</font></span></a>, <a class="txt" href="#E96:1_1_1"><span class="lab"><font color="Green" title="E71">A1109</font></span></a>, <a class="txt" href="#E9:1_1_1_32"><span class="lab"><font color="Green" title="E76">A1114</font></span></a>, <a class="txt" href="#E13:1_1_1_32"><span class="lab"><font color="Green" title="E80">A1118</font></span></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></span>;<br/>
<a NAME="E16:1_1_1_32"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A1119</font></span>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c17">f</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="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:1_1_1_32"><span class="lab"><font color="Green" title="E77">A1115</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E17:1_1_1_32"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> (<font color="Maroon" title="c17">f</font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)</span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c17">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E52:1_1_1"><span class="lab"><font color="Green" title="E44">A655</font></span></a>, <a class="ref" href="jordan1h.html#T25" title="JORDAN1H:th.25">JORDAN1H:25</a></span>;<br/>
<a NAME="E18:1_1_1_32"/><span class="kw">then </span>
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c17">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:1_1_1_32"><span class="lab"><font color="Green" title="E81">A1119</font></span></a>, <a class="txt" href="#E12:1_1_1_32"><span class="lab"><font color="Green" title="E79">A1117</font></span></a>, <a class="ref" href="xboole_1.html#T68" title="XBOOLE_1:th.68">XBOOLE_1:68</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c19">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E20:1_1_1_32"/><span class="lab"><font color="Green" title="E82">A1120</font></span>: 
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">B</font> where <font color="Olive" title="b1">B</font> is   <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> : <font color="Olive" title="b1">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">C</font> </span> } </span> 
 <span class="kw">and </span><br/><a NAME="E21:1_1_1_32"/><span class="lab"><font color="Green" title="E83">A1121</font></span>: 
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c17">f</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c19">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:1_1_1_32"><span class="lab"><font color="Green" title="E78">A1116</font></span></a>, <a class="ref" href="zfmisc_1.html#T80" title="ZFMISC_1:th.80">ZFMISC_1:80</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c20">B</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><span class="kw"> such that </span><br/><a NAME="E23:1_1_1_32"/><span class="lab"><font color="Green" title="E84">A1122</font></span>: 
<font color="Maroon" title="c19">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">B</font>
 <span class="kw">and </span><br/><a NAME="E24:1_1_1_32"/><span class="lab"><font color="Green" title="E85">A1123</font></span>: 
<font color="Maroon" title="c20">B</font> <a href="jordan2c.html#R1" title="JORDAN2C:pred.1">is_inside_component_of</a> <font color="Maroon" title="c1">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:1_1_1_32"><span class="lab"><font color="Green" title="E82">A1120</font></span></a></span>;<br/>
<a NAME="E25:1_1_1_32"/><span class="lab"><font color="Green" title="E86">A1124</font></span>: 
<font color="Maroon" title="c20">B</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1_32"><span class="lab"><font color="Green" title="E85">A1123</font></span></a>, <a class="ref" href="jordan2c.html#D2" title="JORDAN2C:def.2">JORDAN2C:def 2</a></span>;<br/>
<a NAME="E26:1_1_1_32"/>
<font color="Maroon" title="c20">B</font> <a href="connsp_1.html#R3" title="CONNSP_1:pred.3">is_a_component_of</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E24:1_1_1_32"><span class="lab"><font color="Green" title="E85">A1123</font></span></a>, <a class="ref" href="jordan2c.html#D2" title="JORDAN2C:def.2">JORDAN2C:def 2</a></span>;<br/>
<a NAME="E27:1_1_1_32"/><span class="kw">then </span>
 <a href="goboard9.html#K3" title="GOBOARD9:func.3">RightComp</a> <font color="Maroon" title="c17">f</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:1_1_1_32"><span class="lab"><font color="Green" title="E83">A1121</font></span></a>, <a class="txt" href="#E23:1_1_1_32"><span class="lab"><font color="Green" title="E84">A1122</font></span></a>, <a class="txt" href="#E4:1_1_1_32"><span class="lab"><font color="Green" title="E73">A1111</font></span></a>, <a class="txt" href="#E25:1_1_1_32"><span class="lab"><font color="Green" title="E86">A1124</font></span></a>, <a class="ref" href="goboard9.html#T4" title="GOBOARD9:th.4">GOBOARD9:4</a>, <a class="ref" href="jordan2c.html#T12" title="JORDAN2C:th.12">JORDAN2C:12</a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:1_1_1_32"/>
<font color="Maroon" title="c17">f</font> is  <a href="sprect_2.html#V1" title="SPRECT_2:attr.1">clockwise_oriented</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:1_1_1_32"><span class="lab"><font color="Green" title="E75">A1113</font></span></a>, <a class="txt" href="#E5:1_1_1_32"><span class="lab"><font color="Green" title="E74">A1112</font></span></a>, <a class="ref" href="jordan1h.html#T39" title="JORDAN1H:th.39">JORDAN1H:39</a>, <a class="ref" href="jordan1h.html#T41" title="JORDAN1H:th.41">JORDAN1H:41</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="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/>
<span class="kw">take </span>
<font color="Maroon" title="c18">f</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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>) &amp; <font color="Maroon" title="c18">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</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="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) &amp; (  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f</font> holds <br/>( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c18">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c18">f</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c18">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E100:1_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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:1_1_1"><span class="lab"><font color="Green" title="E21">A439</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</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="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</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="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) &amp; (  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f</font> holds <br/>( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c18">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c18">f</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c18">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E101:1_1_1"/>
<font color="Maroon" title="c18">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</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="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E94:1_1_1"><span class="lab"><font color="Green" title="E70">A1108</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c18">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) &amp; (  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f</font> holds <br/>( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c18">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c18">f</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c18">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) ) ) )</span><br/>

<span class="kw">thus </span><a NAME="E102:1_1_1"/>
<font color="Maroon" title="c18">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E96:1_1_1"><span class="lab"><font color="Green" title="E71">A1109</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f</font> holds <br/>( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c18">f</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c18">f</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) &amp; (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<font color="Maroon" title="c18">f</font>,<font color="Olive" title="b1">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>)</span>)</span>) <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> implies <font color="Maroon" title="c18">f</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Olive" title="b1">k</font>, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> (<font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>) ) )</span><br/>

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">m</font> &amp; <font color="Maroon" title="c19">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f</font> implies ( (  <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> &amp;  <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> implies <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>) ) &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> &amp;  <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> implies <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>) ) &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> implies <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>) ) ) )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E103:1_1_1"/><span class="lab"><font color="Green" title="E73">A1125</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c19">m</font>
 <span class="kw">and </span><br/><a NAME="E104:1_1_1"/><span class="lab"><font color="Green" title="E74">A1126</font></span>: 
<font color="Maroon" title="c19">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <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> &amp;  <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> implies <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>) ) &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> &amp;  <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> implies <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>) ) &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> implies <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>) ) )</span><br/>

<a NAME="E105:1_1_1"/><span class="lab"><font color="Green" title="E75">A1127</font></span>: 
<font color="Maroon" title="c7">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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_1"><span class="lab"><font color="Green" title="E31">A564</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E104:1_1_1"><span class="lab"><font color="Green" title="E74">A1126</font></span></a></span>;<br/>
<a NAME="E106:1_1_1"/><span class="lab"><font color="Green" title="E76">A1128</font></span>: 
<font color="Maroon" title="c19">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c19">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/>
<a NAME="E107:1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A1129</font></span>: 
<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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E33:1_1_1"><span class="lab"><font color="Green" title="E31">A564</font></span></a>, <a class="txt" href="#E74:1_1_1"><span class="lab"><font color="Green" title="E58">A709</font></span></a>, <a class="txt" href="#E104:1_1_1"><span class="lab"><font color="Green" title="E74">A1126</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E108:1_1_1"/><span class="lab"><font color="Green" title="E78">A1130</font></span>: 
<font color="Maroon" title="c19">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c18">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E104:1_1_1"><span class="lab"><font color="Green" title="E74">A1126</font></span></a>, <a class="txt" href="#E106:1_1_1"><span class="lab"><font color="Green" title="E76">A1128</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E109:1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A1131</font></span>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">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#K2" title="NAT_1:func.2">+</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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E103:1_1_1"><span class="lab"><font color="Green" title="E73">A1125</font></span></a>, <a class="txt" href="#E107:1_1_1"><span class="lab"><font color="Green" title="E77">A1129</font></span></a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></span>;<br/>
<a NAME="E110:1_1_1"/><span class="lab"><font color="Green" title="E80">A1132</font></span>: 
<font color="Maroon" title="c19">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E103:1_1_1"><span class="lab"><font color="Green" title="E73">A1125</font></span></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/>
<a NAME="E111:1_1_1"/><span class="lab"><font color="Green" title="E81">A1133</font></span>: 
<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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_d.html#K7" title="NAT_D:func.7">-'</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_d.html#T34" title="NAT_D:th.34">NAT_D:34</a></span>;<br/>
<a NAME="E112:1_1_1"/><span class="lab"><font color="Green" title="E82">A1134</font></span>: 
 <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> (<span class="p1">(<span class="default"><font color="Maroon" title="c7">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#K2" title="NAT_1:func.2">+</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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E48:1_1_1"><span class="lab"><font color="Green" title="E41">A652</font></span></a>, <a class="txt" href="#E103:1_1_1"><span class="lab"><font color="Green" title="E73">A1125</font></span></a>, <a class="txt" href="#E108:1_1_1"><span class="lab"><font color="Green" title="E78">A1130</font></span></a>, <a class="txt" href="#E107:1_1_1"><span class="lab"><font color="Green" title="E77">A1129</font></span></a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></span>;<br/>
<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  <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> &amp;  <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> implies <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>) ) &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> implies <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>) ) )</span>
<div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_33"/><span class="lab"><font color="Green" title="E83">A1135</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_33"/><span class="lab"><font color="Green" title="E84">A1136</font></span>: 
 <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/><a NAME="E3:1_1_1_33"/>
<font color="Maroon" title="c7">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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <font color="Maroon" title="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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:1_1_1"><span class="lab"><font color="Green" title="E27">A513</font></span></a>, <a class="txt" href="#E111:1_1_1"><span class="lab"><font color="Green" title="E81">A1133</font></span></a>, <a class="txt" href="#E110:1_1_1"><span class="lab"><font color="Green" title="E80">A1132</font></span></a>, <a class="txt" href="#E109:1_1_1"><span class="lab"><font color="Green" title="E79">A1131</font></span></a>, <a class="txt" href="#E112:1_1_1"><span class="lab"><font color="Green" title="E82">A1134</font></span></a>, <a class="txt" href="#E1:1_1_1_33"><span class="lab"><font color="Green" title="E83">A1135</font></span></a>, <a class="txt" href="#E2:1_1_1_33"><span class="lab"><font color="Green" title="E84">A1136</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_33"/>
<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E103:1_1_1"><span class="lab"><font color="Green" title="E73">A1125</font></span></a>, <a class="txt" href="#E104:1_1_1"><span class="lab"><font color="Green" title="E74">A1126</font></span></a>, <a class="txt" href="#E105:1_1_1"><span class="lab"><font color="Green" title="E75">A1127</font></span></a>, <a class="ref" href="gobrd13.html#T44" title="GOBRD13:th.44">GOBRD13:44</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <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> implies <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>) )</span>
<div class="add"><span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:1_1_1_34"/><span class="lab"><font color="Green" title="E83">A1137</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E2:1_1_1_34"/><span class="lab"><font color="Green" title="E84">A1138</font></span>: 
 <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/><a NAME="E3:1_1_1_34"/>
<font color="Maroon" title="c7">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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <font color="Maroon" title="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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:1_1_1"><span class="lab"><font color="Green" title="E27">A513</font></span></a>, <a class="txt" href="#E111:1_1_1"><span class="lab"><font color="Green" title="E81">A1133</font></span></a>, <a class="txt" href="#E110:1_1_1"><span class="lab"><font color="Green" title="E80">A1132</font></span></a>, <a class="txt" href="#E109:1_1_1"><span class="lab"><font color="Green" title="E79">A1131</font></span></a>, <a class="txt" href="#E112:1_1_1"><span class="lab"><font color="Green" title="E82">A1134</font></span></a>, <a class="txt" href="#E1:1_1_1_34"><span class="lab"><font color="Green" title="E83">A1137</font></span></a>, <a class="txt" href="#E2:1_1_1_34"><span class="lab"><font color="Green" title="E84">A1138</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E4:1_1_1_34"/>
<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E103:1_1_1"><span class="lab"><font color="Green" title="E73">A1125</font></span></a>, <a class="txt" href="#E104:1_1_1"><span class="lab"><font color="Green" title="E74">A1126</font></span></a>, <a class="txt" href="#E105:1_1_1"><span class="lab"><font color="Green" title="E75">A1127</font></span></a>, <a class="ref" href="gobrd13.html#T45" title="GOBRD13:th.45">GOBRD13:45</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E115:1_1_1"/>
 <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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/>

<a NAME="E116:1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c7">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#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <font color="Maroon" title="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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:1_1_1"><span class="lab"><font color="Green" title="E27">A513</font></span></a>, <a class="txt" href="#E111:1_1_1"><span class="lab"><font color="Green" title="E81">A1133</font></span></a>, <a class="txt" href="#E110:1_1_1"><span class="lab"><font color="Green" title="E80">A1132</font></span></a>, <a class="txt" href="#E109:1_1_1"><span class="lab"><font color="Green" title="E79">A1131</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E117:1_1_1"/>
<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E103:1_1_1"><span class="lab"><font color="Green" title="E73">A1125</font></span></a>, <a class="txt" href="#E104:1_1_1"><span class="lab"><font color="Green" title="E74">A1126</font></span></a>, <a class="txt" href="#E105:1_1_1"><span class="lab"><font color="Green" title="E75">A1127</font></span></a>, <a class="ref" href="gobrd13.html#T43" title="GOBRD13:th.43">GOBRD13:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
