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

<b>set </b><font color="Maroon">G</font> =  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">W</font> =  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">E</font> =  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font>;<br/>
<b>set </b><font color="Maroon">N</font> =  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font>;<br/>
<a NAME="E1:33_1_1"/><i><font color="Green">E42</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 3 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 implies  ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp;  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Olive">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 1 implies  ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp;  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Olive">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span> ) ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">a<sub>2</sub></font> is   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> implies for <font color="Olive">f</font> being  <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> holds <br/>( (  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> implies ( ( <font color="Olive">f</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <font color="Olive">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> implies ( (  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Olive">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Olive">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> implies  ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Olive">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Olive">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> implies  ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Olive">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> implies  ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) ) ) &amp; ( (  not <font color="Olive">f</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> or  <a href="gobrd13.html#K3">right_cell</a> <font color="Olive">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> ) implies <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) ) &amp; (  <a href="finseq_1.html#K3">len</a> <font color="Olive">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) ) ) &amp; ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">a<sub>2</sub></font> is  not   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> implies <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) );<br/>
<a NAME="E2:33_1_1"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Olive">x</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:33_1_1_1"/><i><font color="Green">E44</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E3:33_1_1_1"/><i><font color="Green">E45</font></i>: 
(  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or ( <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">x</font> is   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ) or ( <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">x</font> is  not   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ) )
 </span><b>by </b><i><a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>take </b>
<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
;<br/><b>thus </b><a NAME="E2:33_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 <b>by </b><i>, , <a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b>
<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
;<br/><b>thus </b><a NAME="E2:33_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">m</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>]
 <b>by </b><i>, , <a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_1_1_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">x</font> is   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="jordan9.html#T3" target="_self">Th3</a></i>;<br/><b>thus </b><a NAME="E4:33_1_1_1_1_3"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1"/><i><font color="Green">E60</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1"/>
( ( <font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> ) or  not <font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> or  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E62</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><b>then consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E65</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> )
 <b>and </b><br/><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E66</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E7:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E68</font></i>: 
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E8:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E69</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E9:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E70</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E10:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E11:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E72</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E12:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E13:33_1_1_1_1_3_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/><a NAME="E14:33_1_1_1_1_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E74</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:33_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E16:33_1_1_1_1_3_1_1_1_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1"/>
( (  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> ) or (  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> ) or  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E76</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j2</font>;<br/><b>thus </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">f1</font> <a href="gobrd13.html#R1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E78</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E79</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">j1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan9.html#E2:33_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="jordan8.html#T19">JORDAN8:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, , , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_2"/>
( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_3"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_4"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_4"/><i><font color="Green">E81</font></i>: 
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">f1</font>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , <a class="ref" href="jordan9.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i2</font>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1;<br/><b>thus </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1"/>
<font color="Maroon">f1</font> <a href="gobrd13.html#R1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><i><font color="Green">E83</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_1"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2"/>
( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/>
<font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,0 <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="jordan8.html#T20">JORDAN8:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_2"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_3"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_3"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_4"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_4"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">f1</font>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , <a class="ref" href="jordan9.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i2</font>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1;<br/><b>thus </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1"/>
<font color="Maroon">f1</font> <a href="gobrd13.html#R1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><i><font color="Green">E88</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><i><font color="Green">E89</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_1"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_1"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_2"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_2"/><i><font color="Green">E140</font></i>: 
( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"/>
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"/><b>then </b>
( <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jordan8.html#T18">JORDAN8:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, , , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_3"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_4"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_4"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">f1</font>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , <a class="ref" href="jordan9.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"/><i><font color="Green">E141</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>take </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j2</font>;<br/><b>thus </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1"/>
<font color="Maroon">f1</font> <a href="gobrd13.html#R1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><i><font color="Green">E142</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><i><font color="Green">E143</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><b>then </b><i><font color="Green">E144</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_1"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_1"/><i><font color="Green">E145</font></i>: 
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_2"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_2"/>
( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_3"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_3"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,0,<font color="Maroon">j2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jordan8.html#T21">JORDAN8:21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1_4"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">f1</font>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , <a class="ref" href="jordan9.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E146</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E147</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i2</font>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1;<br/><b>thus </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon">f1</font> <a href="gobrd13.html#R3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><i><font color="Green">E148</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><i><font color="Green">E149</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><b>then </b><i><font color="Green">E150</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"/>
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , <a class="ref" href="gobrd13.html#T36">GOBRD13:36</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="jordan8.html#T18">JORDAN8:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, , , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_1"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_2"/>
( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_3"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_4"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_4"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">f1</font>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , <a class="ref" href="jordan9.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E152</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j2</font>;<br/><b>thus </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1"/>
<font color="Maroon">f1</font> <a href="gobrd13.html#R3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><i><font color="Green">E153</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><i><font color="Green">E154</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><b>then </b><i><font color="Green">E155</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_1"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2"/>
( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"/>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , <a class="ref" href="gobrd13.html#T38">GOBRD13:38</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="jordan8.html#T19">JORDAN8:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, , , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_2"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_3"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_3"/><i><font color="Green">E156</font></i>: 
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_4"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_4"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">f1</font>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , <a class="ref" href="jordan9.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"/><i><font color="Green">E157</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j2</font>;<br/><b>thus </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1"/>
<font color="Maroon">f1</font> <a href="gobrd13.html#R3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><i><font color="Green">E158</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><i><font color="Green">E159</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><b>then </b><i><font color="Green">E160</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_1"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_1"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_2"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_2"/><i><font color="Green">E161</font></i>: 
( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , <a class="ref" href="gobrd13.html#T40">GOBRD13:40</a></i>;<br/><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,0,<font color="Maroon">j1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan9.html#E2:33_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="jordan8.html#T21">JORDAN8:21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_3"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_4"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_4"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">f1</font>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , <a class="ref" href="jordan9.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"/><i><font color="Green">E162</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>take </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i2</font>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1;<br/><b>thus </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1"/>
<font color="Maroon">f1</font> <a href="gobrd13.html#R3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><i><font color="Green">E163</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><i><font color="Green">E164</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><b>then </b><i><font color="Green">E165</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_1"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_1"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_2"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_2"/>
( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_3"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_3"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/>
<font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , <a class="ref" href="gobrd13.html#T42">GOBRD13:42</a></i>;<br/><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,0 <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan8.html#T20">JORDAN8:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1_4"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">f1</font>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , <a class="ref" href="jordan9.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3"/><i><font color="Green">E167</font></i>: 
 <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3"/>
 ex <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Olive">y</font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"/><i><font color="Green">E168</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j2</font>;<br/><b>thus </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1"/>
<font color="Maroon">f1</font> <a href="gobrd13.html#R2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><i><font color="Green">E169</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><i><font color="Green">E170</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><b>then </b><i><font color="Green">E171</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , <a class="ref" href="gobrd13.html#T35">GOBRD13:35</a></i>;<br/><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,0,<font color="Maroon">j2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jordan8.html#T21">JORDAN8:21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_1"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_2"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_2"/>
( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_3"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_3"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_4"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_4"/><i><font color="Green">E172</font></i>: 
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">f1</font>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , <a class="ref" href="jordan9.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"/><i><font color="Green">E173</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i2</font>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1;<br/><b>thus </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1"/>
<font color="Maroon">f1</font> <a href="gobrd13.html#R2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><i><font color="Green">E174</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><i><font color="Green">E175</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><b>then </b><i><font color="Green">E178</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_1"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_1"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2"/>
( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"/>
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , <a class="ref" href="gobrd13.html#T37">GOBRD13:37</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jordan8.html#T18">JORDAN8:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, , , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_2"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_3"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_3"/><i><font color="Green">E179</font></i>: 
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_4"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_4"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">f1</font>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , <a class="ref" href="jordan9.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"/><i><font color="Green">E180</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i2</font>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1;<br/><b>thus </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1"/>
<font color="Maroon">f1</font> <a href="gobrd13.html#R2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><i><font color="Green">E181</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><i><font color="Green">E182</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><b>then </b><i><font color="Green">E183</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_1"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_1"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_2"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_2"/><i><font color="Green">E184</font></i>: 
( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/>
<font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , <a class="ref" href="gobrd13.html#T39">GOBRD13:39</a></i>;<br/><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,0 <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan8.html#T20">JORDAN8:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green">E48</font></i></a>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_3"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_4"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_4"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1_4"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">f1</font>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , <a class="ref" href="jordan9.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"/><i><font color="Green">E185</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>take </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j2</font>;<br/><b>thus </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1"/>
<font color="Maroon">f1</font> <a href="gobrd13.html#R2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><i><font color="Green">E186</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><i><font color="Green">E190</font></i>: 
( <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/>

<a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><b>then </b><i><font color="Green">E191</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_1"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_1"/><i><font color="Green">E192</font></i>: 
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2'</font> )
 <b>by </b><i>, , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_2"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_2"/>
( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_2"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_3"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_3"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_3"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4"><b>case </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4"/>
( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"/>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"/><b>then </b>
( <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E5:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , <a class="ref" href="gobrd13.html#T41">GOBRD13:41</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4_1"/>
contradiction
 <b>by </b><i>, <a class="txt" href="jordan9.html#E1:33_1_1_1_1_3"><i><font color="Green">E48</font></i></a>, <a class="ref" href="jordan8.html#T19">JORDAN8:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:33_1_1_1"><i><font color="Green">E45</font></i></a>, , , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E4:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1_4"/>
<font color="Maroon">f1</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2'</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">f1</font>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , <a class="ref" href="jordan9.html#T6" target="_self">Th6</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_1_1_1_2"/><i><font color="Green">E193</font></i>: 
(  not <font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> or  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
;<br/><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>,<font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_3_1_1_2"/><i><font color="Green">E194</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>take </b>
 <a href="xboole_0.html#K1">{}</a> 
;<br/><b>thus </b><a NAME="E2:33_1_1_1_1_3_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>, <a href="xboole_0.html#K1">{}</a> ]
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_1_1_4"/><i><font color="Green">E195</font></i>: 
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">x</font> is  not   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> )
 ;<br/><div class="add"><b>take </b>
 <a href="xboole_0.html#K1">{}</a> 
;<br/><b>thus </b><a NAME="E2:33_1_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">x</font>, <a href="xboole_0.html#K1">{}</a> ]
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">F</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:33_1_1"/><i><font color="Green">E197</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E5:33_1_1"/><i><font color="Green">E198</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>and </b><br/><a NAME="E6:33_1_1"/><i><font color="Green">E199</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">k</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 <b>from </b><i><a class="ref" href="lattice5.html#S1">LATTICE5:sch 1</a>();<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>
<a NAME="E7:33_1_1"/><i><font color="Green">E200</font></i>: 
 <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
;<br/>
<a NAME="E8:33_1_1"/><b>then </b><i><font color="Green">E201</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i/>;<br/>
<a NAME="E9:33_1_1"/><i><font color="Green">E202</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_2"/><i><font color="Green">E203</font></i>: 
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> is   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>

<a NAME="E2:33_1_1_2"/><i><font color="Green">E204</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_2_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_2_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_2_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_2_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_2_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_2_1_3"/><i><font color="Green">E205</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_2_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_2_1_3_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">k</font> as   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_2_1_3_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_2_1_3_1_1_1"/><i><font color="Green">E206</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_2_1_3_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_2_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_2_1_3_1_1_1_1_1"/>
( ( <font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> ) or  not <font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> or  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_2_1_3_1_1_1_1_1_1"/>
( <font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_2_1_3_1_1_1_1_1_1"/><b>then </b>
( (  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> implies  ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> implies  ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> implies  ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_2_1_3_1_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_2_1_3_1_1_1_1_1_2"/><i><font color="Green">E207</font></i>: 
(  not <font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> or  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_2_1_3_1_1_1_1_1_2"/>
<font color="Maroon">f</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> is   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/><b>hence </b><a NAME="E3:33_1_1_2_1_3_1_1_1_1_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_2_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_2_1_3_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_2_1_3_1_1_2"/>
<font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, <a class="ref" href="jordan9.html#T3" target="_self">Th3</a>, <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div><b>end;</b></div>
<a NAME="E10:33_1_1"/><i><font color="Green">E208</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E11:33_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="finseq_2.html#K3">*</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:33_1_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">F</font>
 ;<br/>

<a NAME="E2:33_1_1_3"/><b>then </b>
 ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:33_1_1_3"/><b>then </b>
<font color="Maroon">y</font> is   <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:33_1_1_3"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="finseq_2.html#K3">*</a> 
 <b>by </b><i><a class="ref" href="finseq_1.html#D11">FINSEQ_1:def 11</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <font color="Maroon">F</font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="finseq_2.html#K3">*</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b> <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E13:33_1_1"/><i><font color="Green">E209</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E14:33_1_1"/><i><font color="Green">E210</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_4"/><i><font color="Green">E211</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E2:33_1_1_4"/><i><font color="Green">E212</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>,<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font>,<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_4_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_4_1_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_4_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_4_1_2"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_4_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="finseq_1.html#T61">FINSEQ_1:61</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_4_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_4_1_3"/><i><font color="Green">E213</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>thus </b><a NAME="E2:33_1_1_4_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_4_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_4_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_4_1_3_1_1"/>
( ( <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> ) or  not <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> or  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_4_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_4_1_3_1_1_1"/>
( <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_4_1_3_1_1_1"/><b>then </b>
( (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> implies  ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> implies  ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> implies  ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:33_1_1_4_1_3_1_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_4_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_4_1_3_1_1_2"/>
(  not <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> or  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_4_1_3_1_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,1</span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, <a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, </i>;<br/><b>hence </b><a NAME="E3:33_1_1_4_1_3_1_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_2.html#T19">FINSEQ_2:19</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div><b>end;</b></div>
<a NAME="E15:33_1_1"/><i><font color="Green">E214</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, <a class="txt" href="jordan9.html#E1:33"><i><font color="Green">E41</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; ( for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> holds <br/>(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Olive">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Olive">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> ) ) );<br/>
<a NAME="E16:33_1_1"/><i><font color="Green">E215</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:33_1_1_5"/>
( ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> holds <br/> ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> ) ) &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> holds <br/>for <font color="Olive">m</font>, <font color="Olive">k</font>, <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">m</font>,<font color="Olive">k</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">m</font>,<font color="Olive">k</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">m</font> <a href="real_1.html#K5">-</a> <font color="Olive">i</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">k</font> <a href="real_1.html#K5">-</a> <font color="Olive">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 ) )
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<b>hence </b><a NAME="E2:33_1_1_5"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> 0 <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E3:33_1_1_5"/><i><font color="Green">E216</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> )
 ;<br/>

<a NAME="E4:33_1_1_5"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E5:33_1_1_5"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> 0</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:33_1_1_5"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E7:33_1_1_5"/>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> 0</span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> 0</span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E217</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6"/><i><font color="Green">E218</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:33_1_1_6"/><i><font color="Green">E219</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> holds <br/>(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Olive">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Olive">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>and </b><br/><a NAME="E3:33_1_1_6"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E4:33_1_1_6"/><i><font color="Green">E221</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> )
 <b>and </b><br/><a NAME="E5:33_1_1_6"/><i><font color="Green">E222</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 ;<br/><a NAME="E6:33_1_1_6"/><i><font color="Green">E223</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E7:33_1_1_6"/><b>then </b><i><font color="Green">E224</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E8:33_1_1_6"/><i><font color="Green">E225</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E9:33_1_1_6"/><b>then </b><i><font color="Green">E226</font></i>: 
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E10:33_1_1_6"/><i><font color="Green">E227</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E11:33_1_1_6"/><i><font color="Green">E228</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E12:33_1_1_6"/><i><font color="Green">E229</font></i>: 
( <font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E13:33_1_1_6"/><i><font color="Green">E230</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E14:33_1_1_6"/><i><font color="Green">E231</font></i>: 
( <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_1_6_1"/><i><font color="Green">E232</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_1_1"/>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:33_1_1_6_1_1"/><b>then </b>
( <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:33_1_1_6_1_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_6_1_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E5:33_1_1_6_1_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">j1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , , <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_6_1_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="jordan8.html#T19">JORDAN8:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_6_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_1_6_2"/><i><font color="Green">E233</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_2_1"/>
<font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_6_2_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:33_1_1_6_2_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_6_2_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_6_2_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , , <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/><a NAME="E6:33_1_1_6_2_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,0 <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_6_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan8.html#T20">JORDAN8:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_6_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_1_6_3"/><i><font color="Green">E234</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_3_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_3_1"/>
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:33_1_1_6_3_1"/><b>then </b>
( <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:33_1_1_6_3_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_6_3_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E5:33_1_1_6_3_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , , <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_6_3_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan8.html#T18">JORDAN8:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_6_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_1_6_4"/><i><font color="Green">E235</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_4_1"/>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_6_4_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:33_1_1_6_4_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_6_4_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_6_4_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , , <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/><a NAME="E6:33_1_1_6_4_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,0,<font color="Maroon">j2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_6_4_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="jordan8.html#T21">JORDAN8:21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_6_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6_5"/><i><font color="Green">E236</font></i>: 
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_6_5"/><i><font color="Green">E237</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_5_1"/>
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:33_1_1_6_5_1"/><b>then </b>
( <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:33_1_1_6_5_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_6_5_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E5:33_1_1_6_5_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , , <a class="ref" href="gobrd13.html#T36">GOBRD13:36</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_6_5_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan8.html#T18">JORDAN8:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_6_5"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6_6"/><i><font color="Green">E238</font></i>: 
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_6_6"/><i><font color="Green">E239</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_6_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_6_1"/>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:33_1_1_6_6_1"/><b>then </b>
( <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:33_1_1_6_6_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_6_6_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E5:33_1_1_6_6_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , , <a class="ref" href="gobrd13.html#T38">GOBRD13:38</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_6_6_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="jordan8.html#T19">JORDAN8:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_6_6"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6_7"/><i><font color="Green">E240</font></i>: 
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_6_7"/><i><font color="Green">E241</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_7_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_7_1"/>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_6_7_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:33_1_1_6_7_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_6_7_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_6_7_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , , <a class="ref" href="gobrd13.html#T40">GOBRD13:40</a></i>;<br/><a NAME="E6:33_1_1_6_7_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,0,<font color="Maroon">j1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_6_7_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="jordan8.html#T21">JORDAN8:21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_6_7"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6_8"/><i><font color="Green">E242</font></i>: 
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_6_8"/><i><font color="Green">E243</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_8_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_8_1"/>
<font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_6_8_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:33_1_1_6_8_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_6_8_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_6_8_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , , <a class="ref" href="gobrd13.html#T42">GOBRD13:42</a></i>;<br/><a NAME="E6:33_1_1_6_8_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,0 <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_6_8_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan8.html#T20">JORDAN8:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_6_8"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6_9"/><i><font color="Green">E244</font></i>: 
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_6_9"/><i><font color="Green">E245</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_9_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_9_1"/>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_6_9_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:33_1_1_6_9_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_6_9_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_6_9_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , , <a class="ref" href="gobrd13.html#T35">GOBRD13:35</a></i>;<br/><a NAME="E6:33_1_1_6_9_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,0,<font color="Maroon">j2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_6_9_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="jordan8.html#T21">JORDAN8:21</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_6_9"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6_10"/><i><font color="Green">E246</font></i>: 
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_6_10"/><i><font color="Green">E247</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_10_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_10_1"/>
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:33_1_1_6_10_1"/><b>then </b>
( <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:33_1_1_6_10_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_6_10_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E5:33_1_1_6_10_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , , <a class="ref" href="gobrd13.html#T37">GOBRD13:37</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_6_10_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan8.html#T18">JORDAN8:18</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_6_10"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6_11"/><i><font color="Green">E248</font></i>: 
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_6_11"/><i><font color="Green">E249</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_11_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_11_1"/>
<font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><a NAME="E2:33_1_1_6_11_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><a NAME="E3:33_1_1_6_11_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#T54">BINARITH:54</a></i>;<br/><a NAME="E4:33_1_1_6_11_1"/><b>then </b>
<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:33_1_1_6_11_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , , <a class="ref" href="gobrd13.html#T39">GOBRD13:39</a></i>;<br/><a NAME="E6:33_1_1_6_11_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,0 <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_6_11_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan8.html#T20">JORDAN8:20</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_6_11"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div><div><b>hereby </b>
<div class="add"><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_6_12"/><i><font color="Green">E250</font></i>: 
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_6_12"/><i><font color="Green">E251</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_6_12_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_6_12_1"/>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E2:33_1_1_6_12_1"/><b>then </b>
( <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:33_1_1_6_12_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_6_12_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><a NAME="E5:33_1_1_6_12_1"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , , <a class="ref" href="gobrd13.html#T41">GOBRD13:41</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_6_12_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="jordan8.html#T19">JORDAN8:19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_6_12"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div>
<b>end;</b></div></div><b>end;</b></div>
<a NAME="E18:33_1_1"/><i><font color="Green">E252</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_7"/><i><font color="Green">E253</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:33_1_1_7"/><i><font color="Green">E254</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> holds <br/>(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Olive">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Olive">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/>

<a NAME="E3:33_1_1_7"/><i><font color="Green">E255</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E4:33_1_1_7"/><i><font color="Green">E256</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E5:33_1_1_7"/><i><font color="Green">E257</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E6:33_1_1_7"/><i><font color="Green">E258</font></i>: 
2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 
<b>by </b><i><a class="ref" href="newton.html#T102">NEWTON:102</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_1"/><i><font color="Green">E259</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_1"/><i><font color="Green">E260</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_1"/>
(  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E5:33_1_1_7_1_1"/><i><font color="Green">E261</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a></i>;<br/><a NAME="E6:33_1_1_7_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:33_1_1_7_1_1"/><b>then </b><i><font color="Green">E262</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><div><i><font color="Green">E263</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_1_1"/>
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E2:33_1_1_7_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:33_1_1_7_1_1_1"/><b>then </b>
<font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:33_1_1_7_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_7_1_1_1"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_1_2"/>
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E2:33_1_1_7_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:33_1_1_7_1_1_2"/><b>then </b>
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 1 &amp; 1 <a href="hidden.html#R1">=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_1_2"/>
for <font color="Olive">i1</font>, <font color="Olive">j1</font>, <font color="Olive">i2</font>, <font color="Olive">j2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i1</font>,<font color="Olive">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i2</font>,<font color="Olive">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i1</font>,<font color="Olive">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i2</font>,<font color="Olive">j2</font> holds <br/><span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">i1</font> <a href="real_1.html#K5">-</a> <font color="Olive">i2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Olive">j1</font> <a href="real_1.html#K5">-</a> <font color="Olive">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:33_1_1_7_1_1"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E11:33_1_1_7_1_1"/><i><font color="Green">E264</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E12:33_1_1_7_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E13:33_1_1_7_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E14:33_1_1_7_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E15:33_1_1_7_1_1"/>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_2"/><i><font color="Green">E265</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_2"/><i><font color="Green">E266</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_2"/><i><font color="Green">E267</font></i>: 
(  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E5:33_1_1_7_1_2"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a></i>;<br/><a NAME="E6:33_1_1_7_1_2"/><i><font color="Green">E269</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/><a NAME="E7:33_1_1_7_1_2"/><i><font color="Green">E270</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:33_1_1_7_1_2"/><b>then </b><i><font color="Green">E271</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><div><i><font color="Green">E272</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_2_1"/><i><font color="Green">E273</font></i>: 
<font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E2:33_1_1_7_1_2_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:33_1_1_7_1_2_1"/><b>then </b>
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="cqc_the1.html#T3">CQC_THE1:3</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_2_1"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font> )
 <b>by </b><i>, , , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">l</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_2_2"/><i><font color="Green">E274</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E2:33_1_1_7_1_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> &amp; <font color="Maroon">l</font> <a href="xxreal_0.html#R1">&lt;=</a> 2 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:33_1_1_7_1_2_2"/><b>then </b><i><font color="Green">E275</font></i>: 
( <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> 2 )
 <b>by </b><i><a class="ref" href="cqc_the1.html#T3">CQC_THE1:3</a></i>;<br/><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E4:33_1_1_7_1_2_2"/><i><font color="Green">E276</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">l</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">l</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 ;<br/><a NAME="E5:33_1_1_7_1_2_2"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , , , , , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E6:33_1_1_7_1_2_2"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E7:33_1_1_7_1_2_2"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:33_1_1_7_1_2"/><i><font color="Green">E277</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E12:33_1_1_7_1_2"/><i><font color="Green">E278</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E13:33_1_1_7_1_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E14:33_1_1_7_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E15:33_1_1_7_1_2"/><b>then </b><i><font color="Green">E279</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E16:33_1_1_7_1_2"/><i><font color="Green">E280</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E17:33_1_1_7_1_2"/><b>then </b><i><font color="Green">E281</font></i>: 
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, , , , , , <a class="ref" href="gobrd13.html#D3">GOBRD13:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_2_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_7_1_2_3"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 ;<br/><b>then consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_2_3"/><i><font color="Green">E282</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E6:33_1_1_7_1_2_3"/>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">r</font>,<font color="Olive">s</font></span><a href="euclid.html#K23">]|</a></span> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a>, <font color="Olive">s</font> is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> &amp; <font color="Olive">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">s</font> ) </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, <a class="ref" href="gobrd11.html#T31">GOBRD11:31</a></i>;<br/><b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a>, <font color="Maroon">s</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E8:33_1_1_7_1_2_3"/><i><font color="Green">E283</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">r</font>,<font color="Maroon">s</font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E9:33_1_1_7_1_2_3"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>and </b><br/><a NAME="E10:33_1_1_7_1_2_3"/><i><font color="Green">E284</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">s</font>
 <b>by </b><i/>;<br/><a NAME="E11:33_1_1_7_1_2_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><a NAME="E12:33_1_1_7_1_2_3"/><b>then </b><i><font color="Green">E285</font></i>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/><a NAME="E13:33_1_1_7_1_2_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><b>then </b><i><font color="Green">E286</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
;<br/><a NAME="E15:33_1_1_7_1_2_3"/><i><font color="Green">E287</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 2</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E16:33_1_1_7_1_2_3"/>
 <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan8.html#T12">JORDAN8:12</a></i>;<br/><a NAME="E17:33_1_1_7_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E18:33_1_1_7_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E19:33_1_1_7_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E20:33_1_1_7_1_2_3"/><b>then </b><i><font color="Green">E288</font></i>: 
 <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E21:33_1_1_7_1_2_3"/>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/><b>hence </b><a NAME="E22:33_1_1_7_1_2_3"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E19:33_1_1_7_1_2"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E20:33_1_1_7_1_2"/><i><font color="Green">E289</font></i>: 
 <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="sprect_1.html#T13">SPRECT_1:13</a></i>;<br/><a NAME="E21:33_1_1_7_1_2"/>
 <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , , , , , , , <a class="ref" href="gobrd13.html#D2">GOBRD13:def 2</a></i>;<br/><b>hence </b><a NAME="E22:33_1_1_7_1_2"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3"/><i><font color="Green">E290</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:33_1_1_7_1_3"/><b>then </b>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:33_1_1_7_1_3"/><b>then </b><i><font color="Green">E291</font></i>: 
not <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E4:33_1_1_7_1_3"/><i><font color="Green">E292</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, , <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E5:33_1_1_7_1_3"/><i><font color="Green">E293</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><b>then consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:33_1_1_7_1_3"/><i><font color="Green">E294</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> )
 <b>and </b><br/><a NAME="E8:33_1_1_7_1_3"/><i><font color="Green">E295</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E9:33_1_1_7_1_3"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E10:33_1_1_7_1_3"/><i><font color="Green">E296</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E11:33_1_1_7_1_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/><a NAME="E12:33_1_1_7_1_3"/><b>then </b><i><font color="Green">E297</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:33_1_1_7_1_3"/><i><font color="Green">E298</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E14:33_1_1_7_1_3"/><i><font color="Green">E299</font></i>: 
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E15:33_1_1_7_1_3"/><i><font color="Green">E300</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E16:33_1_1_7_1_3"/><i><font color="Green">E301</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E17:33_1_1_7_1_3"/><i><font color="Green">E302</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E18:33_1_1_7_1_3"/><i><font color="Green">E303</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E19:33_1_1_7_1_3"/><i><font color="Green">E304</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E20:33_1_1_7_1_3"/><i><font color="Green">E305</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> &amp; <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E21:33_1_1_7_1_3"/><i><font color="Green">E306</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E22:33_1_1_7_1_3"/><i><font color="Green">E307</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E23:33_1_1_7_1_3"/><i><font color="Green">E308</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E24:33_1_1_7_1_3"/><i><font color="Green">E309</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> &amp; <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E25:33_1_1_7_1_3"/><i><font color="Green">E310</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E26:33_1_1_7_1_3"/><i><font color="Green">E311</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/><b>thus </b><a NAME="E27:33_1_1_7_1_3"/><i><font color="Green">E312</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_1_3_1_1"/>
( (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> ) or (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> ) or  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_1_1_1"/>
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_3_1_1_1"/><i><font color="Green">E313</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_1_1_1"/><i><font color="Green">E314</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , , </i>;<br/><b>thus </b><a NAME="E5:33_1_1_7_1_3_1_1_1"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<a NAME="E1:33_1_1_7_1_3_1_1_1_1"/><i><font color="Green">E315</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 
<b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E2:33_1_1_7_1_3_1_1_1_1"/><i><font color="Green">E316</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_1_3_1_1_1_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> ) )
 </span><b>by </b><i>, , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , <a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_1_1_1_1"/><i><font color="Green">E317</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_1_1_1_1"/><i><font color="Green">E318</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_1_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_1_1_1_1_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="jordan8.html#T9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_1_1_1_2"/><i><font color="Green">E319</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_1_1_1_2"/><i><font color="Green">E320</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_1_1_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E321</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">j1'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:33_1_1_7_1_3_1_1_1_1_1_2_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j1'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2'</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_7_1_3_1_1_1_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_1_1_1_2"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="jordan8.html#T9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_1_1_1_3"/><i><font color="Green">E322</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_1_1_1_3"/><i><font color="Green">E323</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_1_1_1_3_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_1_1_1_3_1"/><b>then </b>
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_1_1_1_3_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_1_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_1_1_1_3"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="jordan8.html#T9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_1_1_1_4"/><i><font color="Green">E324</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_1_1_1_4"/><i><font color="Green">E325</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_1_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_1_1_1_4_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_1_1_1_4_1"/><b>then </b><i><font color="Green">E326</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_1_1_1_4_1"/><b>then </b>
<font color="Maroon">i1'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:33_1_1_7_1_3_1_1_1_1_1_4_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i1'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2'</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_7_1_3_1_1_1_1_1_4_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_1_1_1_4"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="jordan8.html#T9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_1_1_2"/><i><font color="Green">E327</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_3_1_1_2"/><i><font color="Green">E328</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_1_1_2"/><i><font color="Green">E329</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , , </i>;<br/><b>thus </b><a NAME="E5:33_1_1_7_1_3_1_1_2"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<a NAME="E1:33_1_1_7_1_3_1_1_2_1"/><i><font color="Green">E330</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 
<b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E2:33_1_1_7_1_3_1_1_2_1"/><i><font color="Green">E331</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_1_3_1_1_2_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) )
 </span><b>by </b><i>, , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , <a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_2_1_1_1"/><i><font color="Green">E332</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_2_1_1_1"/><i><font color="Green">E333</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_2_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_2_1_1_1_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_2_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_2_1_1_1"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan8.html#T9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_2_1_1_2"/><i><font color="Green">E334</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_2_1_1_2"/><i><font color="Green">E335</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_2_1_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_2_1_1_2_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_2_1_1_2"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan8.html#T9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_2_1_1_3"/><i><font color="Green">E336</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_2_1_1_3"/><i><font color="Green">E337</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_2_1_1_3_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_2_1_1_3_1"/><b>then </b><i><font color="Green">E338</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_2_1_1_3_1"/><b>then </b>
<font color="Maroon">i1'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:33_1_1_7_1_3_1_1_2_1_1_3_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i1'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2'</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_7_1_3_1_1_2_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_2_1_1_3"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan8.html#T9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_2_1_1_4"/><i><font color="Green">E339</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_2_1_1_4"/><i><font color="Green">E340</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_2_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_2_1_1_4_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_2_1_1_4_1"/><b>then </b><i><font color="Green">E341</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_2_1_1_4_1"/><b>then </b>
<font color="Maroon">j1'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:33_1_1_7_1_3_1_1_2_1_1_4_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j1'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2'</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_7_1_3_1_1_2_1_1_4_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_2_1_1_4"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan8.html#T9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_1_1_3"/><i><font color="Green">E342</font></i>: 
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_3_1_1_3"/><i><font color="Green">E343</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_1_1_3"/><i><font color="Green">E344</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , , </i>;<br/><b>thus </b><a NAME="E5:33_1_1_7_1_3_1_1_3"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">f</font> = <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/>
<a NAME="E1:33_1_1_7_1_3_1_1_3_1"/><i><font color="Green">E345</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 
<b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E2:33_1_1_7_1_3_1_1_3_1"/><i><font color="Green">E346</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_1_3_1_1_3_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> ) )
 </span><b>by </b><i>, , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , <a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_3_1_1_1"/><i><font color="Green">E347</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_3_1_1_1"/><i><font color="Green">E348</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_3_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green">E349</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_3_1_1_1_1"/><b>then </b>
<font color="Maroon">i1'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:33_1_1_7_1_3_1_1_3_1_1_1_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i1'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2'</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_7_1_3_1_1_3_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_3_1_1_1"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan8.html#T9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_3_1_1_2"/><i><font color="Green">E350</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_3_1_1_2"/><i><font color="Green">E351</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_3_1_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_3_1_1_2_1"/><b>then </b>
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_3_1_1_2_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_3_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_3_1_1_2"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan8.html#T9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_3_1_1_3"/><i><font color="Green">E352</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_3_1_1_3"/><i><font color="Green">E353</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_3_1_1_3_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_3_1_1_3_1"/><b>then </b><i><font color="Green">E354</font></i>: 
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_3_1_1_3_1"/><b>then </b>
<font color="Maroon">j1'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:33_1_1_7_1_3_1_1_3_1_1_3_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j1'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2'</font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_7_1_3_1_1_3_1_1_3_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="ref" href="uniform1.html#T13">UNIFORM1:13</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_3_1_1_3"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan8.html#T9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_1_1_3_1_1_4"/><i><font color="Green">E355</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_1_1_3_1_1_4"/><i><font color="Green">E356</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_1_1_3_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:33_1_1_7_1_3_1_1_3_1_1_4_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font> )
 ;<br/><a NAME="E2:33_1_1_7_1_3_1_1_3_1_1_4_1"/><b>then </b>
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_1_1_3_1_1_4_1"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp;  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_3_1_1_4_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i1'</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j2'</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j1'</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_1_7_1_3_1_1_3_1_1_4"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan8.html#T9">JORDAN8:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div><b>end;</b></div><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E28:33_1_1_7_1_3"/><i><font color="Green">E357</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_1_3_2_1"/>
( <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> or <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:33_1_1_7_1_3_2_1_1"/><b>then </b><i><font color="Green">E358</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T4" target="_self">Th4</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_2_1_1"/><i><font color="Green">E359</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E4:33_1_1_7_1_3_2_1_1"/><i><font color="Green">E360</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><b>thus </b><a NAME="E5:33_1_1_7_1_3_2_1_1"/>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1"/>
( (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> ) or (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> ) or  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_1"/><i><font color="Green">E361</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_1"/><i><font color="Green">E362</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_1"/><i><font color="Green">E363</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , , </i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_1"/><i><font color="Green">E364</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> ) )
 </span><b>by </b><i>, , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , <a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_1"/><i><font color="Green">E365</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_1_1_1_1"/><i><font color="Green">E366</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_1_1_1_1"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j2</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="gobrd13.html#T36">GOBRD13:36</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_1_1_1_1"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> &amp;  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , <a class="ref" href="binarith.html#T39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_1_3_2_1_1_1_1_1_1_1_1"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_1_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_2"/><i><font color="Green">E367</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_1_1_1_2"/><i><font color="Green">E368</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_1_1_1_2"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="gobrd13.html#T38">GOBRD13:38</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_1_1_1_2"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , , <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_1_1_1_2"/>
( <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp;  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , <a class="ref" href="binarith.html#T39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_1_3_2_1_1_1_1_1_1_1_2"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_3"/><i><font color="Green">E369</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_1_1_1_3"/><i><font color="Green">E370</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_1_1_1_3"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="gobrd13.html#T40">GOBRD13:40</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_1_1_1_3"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_1_1_1_3"/>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_1_3_2_1_1_1_1_1_1_1_3"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_1_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_1_1_1_4"/><i><font color="Green">E371</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_1_1_1_4"/><i><font color="Green">E372</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_1_1_1_4"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="gobrd13.html#T42">GOBRD13:42</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_1_1_1_4"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , , <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_1_1_1_4"/>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_1_3_2_1_1_1_1_1_1_1_4"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:33_1_1_7_1_3_2_1_1_1_1_1"/>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_2"/><i><font color="Green">E373</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_2"/><i><font color="Green">E374</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_2"/><i><font color="Green">E375</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , , </i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_2"/><i><font color="Green">E376</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) )
 </span><b>by </b><i>, , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , <a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_2_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_1"/><i><font color="Green">E377</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_2_1_1_1"/><i><font color="Green">E378</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="gobrd13.html#T35">GOBRD13:35</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j2</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="gobrd13.html#T36">GOBRD13:36</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_1_3_2_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_2_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_2"/><i><font color="Green">E379</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_2_1_1_2"/><i><font color="Green">E380</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="gobrd13.html#T37">GOBRD13:37</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="gobrd13.html#T38">GOBRD13:38</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_1_3_2_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_2_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_3"/><i><font color="Green">E381</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_2_1_1_3"/><i><font color="Green">E382</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="gobrd13.html#T39">GOBRD13:39</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , , <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="gobrd13.html#T40">GOBRD13:40</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_1_3_2_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , , <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_2_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_2_1_1_4"/><i><font color="Green">E383</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_2_1_1_4"/><i><font color="Green">E384</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="gobrd13.html#T41">GOBRD13:41</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , , <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="gobrd13.html#T42">GOBRD13:42</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_1_3_2_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , , <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:33_1_1_7_1_3_2_1_1_1_1_2"/>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_3"/><i><font color="Green">E385</font></i>: 
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_3"/><i><font color="Green">E386</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_3"/><i><font color="Green">E387</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , , </i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_3"/><i><font color="Green">E388</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> ) )
 </span><b>by </b><i>, , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , <a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_3_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_1"/><i><font color="Green">E389</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_3_1_1_1"/><i><font color="Green">E390</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_3_1_1_1"/>
( <font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> &amp;  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , <a class="ref" href="binarith.html#T39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_3_1_1_1"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , , <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_3_1_1_1"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="gobrd13.html#T35">GOBRD13:35</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_1_3_2_1_1_1_1_3_1_1_1"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , , <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_3_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_2"/><i><font color="Green">E391</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_3_1_1_2"/><i><font color="Green">E392</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_3_1_1_2"/>
( <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp;  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , <a class="ref" href="binarith.html#T39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_3_1_1_2"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_3_1_1_2"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="gobrd13.html#T37">GOBRD13:37</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_1_3_2_1_1_1_1_3_1_1_2"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_3_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_3"/><i><font color="Green">E393</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_3_1_1_3"/><i><font color="Green">E394</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_3_1_1_3"/>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_3_1_1_3"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , , <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_3_1_1_3"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="gobrd13.html#T39">GOBRD13:39</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_1_3_2_1_1_1_1_3_1_1_3"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , , <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_1_1_1_3_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:33_1_1_7_1_3_2_1_1_1_1_3_1_1_4"/><i><font color="Green">E395</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:33_1_1_7_1_3_2_1_1_1_1_3_1_1_4"/><i><font color="Green">E396</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<br/><div class="add"><a NAME="E3:33_1_1_7_1_3_2_1_1_1_1_3_1_1_4"/>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_7_1_3_2_1_1_1_1_3_1_1_4"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>;<br/><a NAME="E5:33_1_1_7_1_3_2_1_1_1_1_3_1_1_4"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="gobrd13.html#T41">GOBRD13:41</a></i>;<br/><b>hence </b><a NAME="E6:33_1_1_7_1_3_2_1_1_1_1_3_1_1_4"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , , , , , , <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:33_1_1_7_1_3_2_1_1_1_1_3"/>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_2"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:33_1_1_7_1_3_2_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:33_1_1_7_1_3_2_1_2"/><b>then </b><i><font color="Green">E397</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:33_1_1_7_1_3_2_1_2"/><i><font color="Green">E398</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> )
 <b>and </b><br/><a NAME="E6:33_1_1_7_1_3_2_1_2"/><i><font color="Green">E399</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E7:33_1_1_7_1_3_2_1_2"/><i><font color="Green">E400</font></i>: 
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E8:33_1_1_7_1_3_2_1_2"/><i><font color="Green">E401</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E9:33_1_1_7_1_3_2_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:33_1_1_7_1_3_2_1_2"/><b>then </b><i><font color="Green">E402</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:33_1_1_7_1_3_2_1_2"/><i><font color="Green">E403</font></i>: 
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , </i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_1_3_2_1_2_1_1"/>
( (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> ) or (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> ) or  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_2_1_1_1"/>
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_3_2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_2_1_2_1_1_1"/><i><font color="Green">E404</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , , </i>;<br/><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i</font>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j</font>;<br/><b>thus </b><a NAME="E5:33_1_1_7_1_3_2_1_2_1_1_1"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_2_1_1_2"/>
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_3_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_2_1_2_1_1_2"/><i><font color="Green">E405</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , , </i>;<br/><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i</font>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j</font>;<br/><b>thus </b><a NAME="E5:33_1_1_7_1_3_2_1_2_1_1_2"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_2_1_1_3"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_7_1_3_2_1_2_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:33_1_1_7_1_3_2_1_2_1_1_3"/><i><font color="Green">E406</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , , </i>;<br/><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i</font>;<br/><b>take </b><font color="Maroon">j</font> = <font color="Maroon">j</font>;<br/><b>thus </b><a NAME="E5:33_1_1_7_1_3_2_1_2_1_1_3"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E14:33_1_1_7_1_3_2_1_2"/><i><font color="Green">E407</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/><a NAME="E15:33_1_1_7_1_3_2_1_2"/><i><font color="Green">E408</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 <b>by </b><i>, , , , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_7_1_3_2_1_2_2_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_2_2_1_1"/><i><font color="Green">E409</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_7_1_3_2_1_2_2_1_1"/><b>then </b>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font> )
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_7_1_3_2_1_2_2_1_1"/>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , , , , , <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_2_2_1_2"/><i><font color="Green">E410</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_7_1_3_2_1_2_2_1_2"/><b>then </b>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_7_1_3_2_1_2_2_1_2"/>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , , , , , <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_2_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_2_2_1_3"/><i><font color="Green">E411</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_7_1_3_2_1_2_2_1_3"/><b>then </b>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_7_1_3_2_1_2_2_1_3"/>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , , , , , <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_7_1_3_2_1_2_2_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_7_1_3_2_1_2_2_1_4"/><i><font color="Green">E412</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_7_1_3_2_1_2_2_1_4"/><b>then </b>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> )
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_7_1_3_2_1_2_2_1_4"/>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , , , , , <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E17:33_1_1_7_1_3_2_1_2"/>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E30:33_1_1_7_1_3"/>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E19:33_1_1"/><i><font color="Green">E413</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="jordan9.html#E2:33_1_1"><i><font color="Green">E43</font></i></a>, );<br/></i>
<a NAME="E20:33_1_1"/><i><font color="Green">E414</font></i>: 
for <font color="Olive">k</font>, <font color="Olive">i1</font>, <font color="Olive">i2</font>, <font color="Olive">j1</font>, <font color="Olive">j2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i1</font>,<font color="Olive">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i1</font>,<font color="Olive">j1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i2</font>,<font color="Olive">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i2</font>,<font color="Olive">j2</font> holds <br/>( (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> implies ( <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; ( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; ( <font color="Olive">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i2</font>,<span class="p2">(<span class="default"><font color="Olive">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i2</font>,<span class="p3">(<span class="default"><font color="Olive">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; ( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i2</font>,<span class="p2">(<span class="default"><font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i2</font>,<span class="p3">(<span class="default"><font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; ( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Olive">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Olive">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) ) ) &amp; (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> implies ( <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; ( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i2</font>,<span class="p2">(<span class="default"><font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i2</font>,<span class="p3">(<span class="default"><font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; ( <font color="Olive">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; ( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Olive">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Olive">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; ( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i2</font>,<span class="p2">(<span class="default"><font color="Olive">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i2</font>,<span class="p3">(<span class="default"><font color="Olive">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) ) ) &amp; (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> implies ( <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; ( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Olive">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Olive">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; ( <font color="Olive">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i2</font>,<span class="p2">(<span class="default"><font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i2</font>,<span class="p3">(<span class="default"><font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; ( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i2</font>,<span class="p2">(<span class="default"><font color="Olive">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i2</font>,<span class="p3">(<span class="default"><font color="Olive">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; ( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_8"/><i><font color="Green">E415</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:33_1_1_8"/><i><font color="Green">E416</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> )
 <b>and </b><br/><a NAME="E3:33_1_1_8"/><i><font color="Green">E417</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 ;<br/>

<a NAME="E4:33_1_1_8"/><i><font color="Green">E418</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/>
<a NAME="E5:33_1_1_8"/><i><font color="Green">E419</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:33_1_1_8"/><i><font color="Green">E420</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E7:33_1_1_8"/><b>then </b><i><font color="Green">E421</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/>
<a NAME="E8:33_1_1_8"/><i><font color="Green">E422</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E9:33_1_1_8"/><b>then </b><i><font color="Green">E423</font></i>: 
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:33_1_1_8"/><i><font color="Green">E424</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:33_1_1_8"/><i><font color="Green">E425</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E12:33_1_1_8"/><i><font color="Green">E426</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j2</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_1_8_1"/>
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 ;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_8_1"/><i><font color="Green">E427</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:33_1_1_8_1"/><i><font color="Green">E428</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , , </i>;<br/><a NAME="E5:33_1_1_8_1"/><i><font color="Green">E429</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E6:33_1_1_8_1"/><i><font color="Green">E430</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/><b>thus </b><a NAME="E7:33_1_1_8_1"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E8:33_1_1_8_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , , , , , , , <a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/><b>thus </b><a NAME="E9:33_1_1_8_1"/>
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , , , , , , , <a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/><b>thus </b><a NAME="E10:33_1_1_8_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , , , , , , , <a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/><b>thus </b><a NAME="E11:33_1_1_8_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , , , , , , , <a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_1_8_2"/>
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/><b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_8_2"/><i><font color="Green">E431</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E4:33_1_1_8_2"/><i><font color="Green">E432</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , , </i>;<br/><a NAME="E5:33_1_1_8_2"/><i><font color="Green">E433</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E6:33_1_1_8_2"/><i><font color="Green">E434</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/><b>thus </b><a NAME="E7:33_1_1_8_2"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E8:33_1_1_8_2"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , , , , , , , <a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E9:33_1_1_8_2"/>
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , , , , , , , <a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E10:33_1_1_8_2"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , , , , , , , <a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E11:33_1_1_8_2"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , , , , , , , <a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E15:33_1_1_8"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E17:33_1_1_8"/><i><font color="Green">E435</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> <a href="gobrd13.html#R2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E18:33_1_1_8"/><i><font color="Green">E436</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a>, , , , </i>;<br/>
<a NAME="E19:33_1_1_8"/><i><font color="Green">E437</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 
<b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E20:33_1_1_8"/><i><font color="Green">E438</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/>
<b>thus </b><a NAME="E21:33_1_1_8"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E22:33_1_1_8"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , , , , , , , <a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/>

<b>thus </b><a NAME="E23:33_1_1_8"/>
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , , , , , , , <a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/>

<b>thus </b><a NAME="E24:33_1_1_8"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , , , , , , , <a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/>

<b>thus </b><a NAME="E25:33_1_1_8"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , , , , , , , <a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E21:33_1_1"/><i><font color="Green">E439</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:33_1_1_9"/><i><font color="Green">E440</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:33_1_1_9"/><i><font color="Green">E441</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E3:33_1_1_9"/><i><font color="Green">E442</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 3 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/>
<a NAME="E4:33_1_1_9"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 4
 
<b>by </b><i><a class="ref" href="jordan8.html#T13">JORDAN8:13</a></i>;<br/>
<a NAME="E5:33_1_1_9"/><b>then </b><i><font color="Green">E443</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_9_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_1"/><b>then </b><i><font color="Green">E444</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:33_1_1_9_1_1"/><i><font color="Green">E445</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E5:33_1_1_9_1_1"/>
(  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E6:33_1_1_9_1_1"/><i><font color="Green">E446</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a></i>;<br/><b>take </b>
<font color="Maroon">i</font>
;<br/><b>take </b><font color="Maroon">j</font> =  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>;<br/><a NAME="E7:33_1_1_9_1_1"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E8:33_1_1_9_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E9:33_1_1_9_1_1"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, , , <a class="ref" href="finseq_1.html#T47">FINSEQ_1:47</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_2"/><i><font color="Green">E447</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_9_1_2"/><i><font color="Green">E448</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:33_1_1_9_1_2"/><i><font color="Green">E449</font></i>: 
(  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E5:33_1_1_9_1_2"/><i><font color="Green">E450</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a></i>;<br/><b>consider </b><font color="Maroon">i'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:33_1_1_9_1_2"/><i><font color="Green">E451</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i'</font> &amp; <font color="Maroon">i'</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp;  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i'</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i'</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E8:33_1_1_9_1_2"/><i><font color="Green">E452</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i'</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a></i>;<br/><a NAME="E9:33_1_1_9_1_2"/><i><font color="Green">E453</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i'</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, </i>;<br/><b>take </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
;<br/><b>take </b><font color="Maroon">j</font> =  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>;<br/><a NAME="E10:33_1_1_9_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E11:33_1_1_9_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><b>thus </b><a NAME="E12:33_1_1_9_1_2"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, , , , <a class="ref" href="finseq_1.html#D9">FINSEQ_1:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3"/><i><font color="Green">E454</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3"/><b>then </b><i><font color="Green">E455</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E3:33_1_1_9_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, , <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><b>then consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:33_1_1_9_1_3"/><i><font color="Green">E456</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> )
 <b>and </b><br/><a NAME="E6:33_1_1_9_1_3"/><i><font color="Green">E457</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E7:33_1_1_9_1_3"/><i><font color="Green">E458</font></i>: 
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_9_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_9_1_3_1_1"/>
( (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> ) or (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> ) or  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_1"/><i><font color="Green">E459</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_9_1_3_1_1_1_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_1_1_1_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_1_1_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_1_3_1_1_1_1_1_1"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_1_1_1_2"/>
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_1_1_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_1_3_1_1_1_1_1_2"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_1_1_1_3"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_1_1_1_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_1_3_1_1_1_1_1_3"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_1_1_1_4"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_1_1_1_4"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_1_3_1_1_1_1_1_4"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_9_1_3_1_1_1"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_2"/><i><font color="Green">E460</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_9_1_3_1_1_2_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_2_1_1_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_2_1_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_1_3_1_1_2_1_1_1"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_2_1_1_2"/>
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_2_1_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_1_3_1_1_2_1_1_2"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_2_1_1_3"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_2_1_1_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_1_3_1_1_2_1_1_3"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_2_1_1_4"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_2_1_1_4"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_1_3_1_1_2_1_1_4"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_9_1_3_1_1_2"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_3"/><i><font color="Green">E461</font></i>: 
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_9_1_3_1_1_3_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_3_1_1_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_3_1_1_1"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_1_3_1_1_3_1_1_1"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_3_1_1_2"/>
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_3_1_1_2"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_1_3_1_1_3_1_1_2"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_3_1_1_3"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_3_1_1_3"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_1_3_1_1_3_1_1_3"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_9_1_3_1_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_9_1_3_1_1_3_1_1_4"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_9_1_3_1_1_3_1_1_4"/><b>then </b>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, , , , </i>;<br/><b>hence </b><a NAME="E3:33_1_1_9_1_3_1_1_3_1_1_4"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_9_1_3_1_1_3"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E9:33_1_1_9_1_3"/>
 ex <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i</font>,<font color="Olive">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<font color="Olive">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">m</font>;<br/>
<a NAME="E22:33_1_1"/><i><font color="Green">E462</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_10"/><i><font color="Green">E463</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:33_1_1_10"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> <a href="finseq_1.html#K17">|</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font>
 
<b>by </b><i><a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E3:33_1_1_10"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E23:33_1_1"/><i><font color="Green">E464</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_11"/><i><font color="Green">E465</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">m</font>
 ;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E2:33_1_1_11"/><i><font color="Green">E466</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_11_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_11_1"/>
( <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_11_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_11_1_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:33_1_1_11_1_1"/><b>then </b><i><font color="Green">E467</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:33_1_1_11_1_1"/><i><font color="Green">E468</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/><b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:33_1_1_11_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:33_1_1_11_1_1"/><i><font color="Green">E469</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i/>;<br/><a NAME="E7:33_1_1_11_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , <a class="ref" href="finseq_5.html#T25">FINSEQ_5:25</a></i>;<br/><b>hence </b><a NAME="E8:33_1_1_11_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_11_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_11_1_2"/><i><font color="Green">E470</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:33_1_1_11_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_11_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E24:33_1_1"/><i><font color="Green">E471</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="jordan9.html#T4" target="_self">Th4</a>, <a class="ref" href="jordan9.html#T5" target="_self">Th5</a>);<br/></i>
<a NAME="E25:33_1_1"/><i><font color="Green">E472</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 holds <br/>( (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> implies <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R1">turns_right</a> <font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> ) &amp; (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> implies <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R3">goes_straight</a> <font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> ) &amp; (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> implies <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R2">turns_left</a> <font color="Olive">k</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_12"/><i><font color="Green">E473</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/>

<a NAME="E2:33_1_1_12"/><i><font color="Green">E474</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:33_1_1_12"/><i><font color="Green">E475</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E4:33_1_1_12"/><b>then </b><i><font color="Green">E476</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/>
<a NAME="E5:33_1_1_12"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E6:33_1_1_12"/><b>then </b>
 ex <font color="Olive">i1</font>, <font color="Olive">j1</font>, <font color="Olive">i2</font>, <font color="Olive">j2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i1</font>,<font color="Olive">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i1</font>,<font color="Olive">j1</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i2</font>,<font color="Olive">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i2</font>,<font color="Olive">j2</font> &amp; ( ( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> ) or ( <font color="Olive">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> ) or ( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> ) or ( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1 ) ) )
 
<b>by </b><i>, , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/>
<b>hence </b><a NAME="E7:33_1_1_12"/>
( (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> implies <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R1">turns_right</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> ) &amp; (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> implies <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R3">goes_straight</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> ) &amp; (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> implies <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R2">turns_left</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> ) )
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>6</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is <a href="topreal1.html#V2">unfolded</a>;<br/>
<a NAME="E26:33_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E27:33_1_1"/><b>then </b><i><font color="Green">E477</font></i>: 
<font color="Maroon">S<sub>6</sub></font>[0]
 
<b>by </b><i><a class="ref" href="sppol_2.html#T27">SPPOL_2:27</a></i>;<br/>
<a NAME="E28:33_1_1"/><i><font color="Green">E478</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>6</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_13"/><i><font color="Green">E479</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font> is <a href="topreal1.html#V2">unfolded</a>
 ;<br/>

<a NAME="E2:33_1_1_13"/><i><font color="Green">E480</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_13_1"/>
( <font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 1 or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:33_1_1_13_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 2
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_13_1_1"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i><a class="ref" href="sppol_2.html#T27">SPPOL_2:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2"/><i><font color="Green">E481</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon">m</font> = <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1;<br/><a NAME="E2:33_1_1_13_1_2"/><i><font color="Green">E482</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E3:33_1_1_13_1_2"/><i><font color="Green">E483</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E4:33_1_1_13_1_2"/><i><font color="Green">E484</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:33_1_1_13_1_2"/><i><font color="Green">E485</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> )
 <b>and </b><br/><a NAME="E7:33_1_1_13_1_2"/><i><font color="Green">E486</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E8:33_1_1_13_1_2"/><i><font color="Green">E487</font></i>: 
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E9:33_1_1_13_1_2"/><i><font color="Green">E488</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E10:33_1_1_13_1_2"/><i><font color="Green">E489</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E11:33_1_1_13_1_2"/><b>then </b><i><font color="Green">E490</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E12:33_1_1_13_1_2"/><i><font color="Green">E491</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E13:33_1_1_13_1_2"/><i><font color="Green">E492</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span>
 <b>by </b><i>, , , , , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_13_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_13_1_2_1_1"/>
( (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> ) or (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> ) or  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_1"/><i><font color="Green">E493</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_13_1_2_1_1_1_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_1_1_1_1"/><i><font color="Green">E494</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E495</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , </i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , , , <a class="ref" href="goboard7.html#T17">GOBOARD7:17</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_13_1_2_1_1_1_1_1_1"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i>, , , , <a class="ref" href="sppol_2.html#T31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_1_1_1_2"/><i><font color="Green">E496</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E497</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , </i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_1_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 &amp;  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_1_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , , , , <a class="ref" href="goboard7.html#T18">GOBOARD7:18</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_13_1_2_1_1_1_1_1_2"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i>, , , , <a class="ref" href="sppol_2.html#T31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_1_1_1_3"/><i><font color="Green">E498</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E499</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , </i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_1_1_1_3"/><b>then </b>
<font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , , , <a class="ref" href="goboard7.html#T19">GOBOARD7:19</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_13_1_2_1_1_1_1_1_3"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i>, , , , <a class="ref" href="sppol_2.html#T31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_1_1_1_4"/><i><font color="Green">E500</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_1_1_1_4"/><b>then </b><i><font color="Green">E501</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , </i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_1_1_1_4"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_1_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , , , , <a class="ref" href="goboard7.html#T20">GOBOARD7:20</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_13_1_2_1_1_1_1_1_4"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i>, , , , <a class="ref" href="sppol_2.html#T31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_13_1_2_1_1_1"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_2"/><i><font color="Green">E502</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_13_1_2_1_1_2_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_2_1_1_1"/><i><font color="Green">E503</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E504</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , </i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_2_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , , <a class="ref" href="goboard7.html#T15">GOBOARD7:15</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_13_1_2_1_1_2_1_1_1"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i>, , , , <a class="ref" href="sppol_2.html#T31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_2_1_1_2"/><i><font color="Green">E505</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E506</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , </i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_2_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , , <a class="ref" href="goboard7.html#T16">GOBOARD7:16</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_13_1_2_1_1_2_1_1_2"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i>, , , , <a class="ref" href="sppol_2.html#T31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_2_1_1_3"/><i><font color="Green">E507</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_2_1_1_3"/><b>then </b><i><font color="Green">E508</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , </i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_2_1_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_2_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , , , <a class="ref" href="goboard7.html#T16">GOBOARD7:16</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_13_1_2_1_1_2_1_1_3"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i>, , , , <a class="ref" href="sppol_2.html#T31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_2_1_1_4"/><i><font color="Green">E509</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_2_1_1_4"/><b>then </b><i><font color="Green">E510</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , </i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_2_1_1_4"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_2_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , , , <a class="ref" href="goboard7.html#T15">GOBOARD7:15</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_13_1_2_1_1_2_1_1_4"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i>, , , , <a class="ref" href="sppol_2.html#T31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_13_1_2_1_1_2"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_3"/><i><font color="Green">E511</font></i>: 
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_13_1_2_1_1_3_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_3_1_1_1"/><i><font color="Green">E512</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E513</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , </i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_3_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1 &amp;  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_3_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , , , , <a class="ref" href="goboard7.html#T18">GOBOARD7:18</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_13_1_2_1_1_3_1_1_1"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i>, , , , <a class="ref" href="sppol_2.html#T31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_3_1_1_2"/><i><font color="Green">E514</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E515</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , </i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_3_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_3_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , , , <a class="ref" href="goboard7.html#T20">GOBOARD7:20</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_13_1_2_1_1_3_1_1_2"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i>, , , , <a class="ref" href="sppol_2.html#T31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_3_1_1_3"/><i><font color="Green">E516</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_3_1_1_3"/><b>then </b><i><font color="Green">E517</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , </i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_3_1_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1 &amp;  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_3_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p3">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , , , , <a class="ref" href="goboard7.html#T17">GOBOARD7:17</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_13_1_2_1_1_3_1_1_3"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i>, , , , <a class="ref" href="sppol_2.html#T31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_13_1_2_1_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_13_1_2_1_1_3_1_1_4"/><i><font color="Green">E518</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_13_1_2_1_1_3_1_1_4"/><b>then </b><i><font color="Green">E519</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, , , , </i>;<br/><a NAME="E3:33_1_1_13_1_2_1_1_3_1_1_4"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp;  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:33_1_1_13_1_2_1_1_3_1_1_4"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">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">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i>, , , , , <a class="ref" href="goboard7.html#T19">GOBOARD7:19</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1_13_1_2_1_1_3_1_1_4"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 <b>by </b><i>, , , , <a class="ref" href="sppol_2.html#T31">SPPOL_2:31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_13_1_2_1_1_3"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E15:33_1_1_13_1_2"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="topreal1.html#V2">unfolded</a>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E29:33_1_1"/><i><font color="Green">E520</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>defpred </b><font color="Maroon">S<sub>7</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp;  ex <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span></span>)</span> ) );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_14"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_14"/><i><font color="Green">E522</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> &amp; <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span></span>)</span>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>8</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a>;<br/><a NAME="E2:33_1_1_14"/><i><font color="Green">E523</font></i>: 
<font color="Maroon">S<sub>8</sub></font>[0]
 <b>by </b><i/>;<br/><a NAME="E3:33_1_1_14"/><i><font color="Green">E524</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>8</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>8</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_14_1"/><i><font color="Green">E525</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_14_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_14_1_1"/><i><font color="Green">E526</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E2:33_1_1_14_1_1"/><i><font color="Green">E527</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font>
 ;<br/><b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:33_1_1_14_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i</font>,<font color="Maroon">j</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E5:33_1_1_14_1_1"/><i><font color="Green">E528</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<font color="Maroon">j</font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i/>;<br/><a NAME="E6:33_1_1_14_1_1"/><i><font color="Green">E529</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:33_1_1_14_1_1"/><i><font color="Green">E530</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E8:33_1_1_14_1_1"/><i><font color="Green">E531</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_14_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_14_1_1_1"/>
( ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> ) or ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> ) or ( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i>, , <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_14_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_14_1_1_1_1"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_14_1_1_1_1"/><b>then </b><i><font color="Green">E532</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:33_1_1_14_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> )
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_14_1_1_1_1"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , <a class="ref" href="partfun2.html#T17">PARTFUN2:17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_14_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_14_1_1_1_2"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_14_1_1_1_2"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_14_1_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_14_1_1_1_3"/>
( <font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_14_1_1_1_3"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_14_1_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_14_1_1_1_4"/>
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_14_1_1_1_4"/>
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:33_1_1_14_1"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="partfun2.html#T16">PARTFUN2:16</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:33_1_1_14"/><i><font color="Green">E533</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>8</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="ref" href="jordan9.html#T7" target="_self">Th7</a>, );<br/></i><a NAME="E5:33_1_1_14"/><i><font color="Green">E534</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds   <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">k</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_14_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:33_1_1_14_2"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i/>;<br/>
<b>hence </b> <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>
<br/>.= 
<font color="Maroon">k</font>
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">k</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/><a NAME="E6:33_1_1_14"/><i><font color="Green">E535</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:33_1_1_14"/><i><font color="Green">E536</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K2">Values</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="gobrd13.html#T8">GOBRD13:8</a></i>;<br/><a NAME="E8:33_1_1_14"/>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E9:33_1_1_14"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p4">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p5">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="gobrd13.html#K2">Values</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="gobrd13.html#T9">GOBRD13:9</a></i>;<br/><a NAME="E10:33_1_1_14"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K2">Values</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/><a NAME="E11:33_1_1_14"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p5">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="nat_1.html#K2">*</a> <span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E12:33_1_1_14"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<a NAME="E31:33_1_1"/><b>then </b><i><font color="Green">E537</font></i>: 
 ex <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive">k</font>]
 
;<br/>
<b>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E33:33_1_1"/><i><font color="Green">E538</font></i>: 
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon">k</font>]
 <b>and </b><br/><a NAME="E34:33_1_1"/><i><font color="Green">E539</font></i>: 
for <font color="Olive">l</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive">l</font>] holds <br/><font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">l</font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>();<br/></i>
<b>consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E36:33_1_1"/><i><font color="Green">E540</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>and </b><br/><a NAME="E37:33_1_1"/><i><font color="Green">E541</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 <b>and </b><br/><a NAME="E38:33_1_1"/><i><font color="Green">E542</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E39:33_1_1"/><i><font color="Green">E543</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E40:33_1_1"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="jordan9.html#T7" target="_self">Th7</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E42:33_1_1"/><i><font color="Green">E544</font></i>: 
<font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E43:33_1_1"/><i><font color="Green">E545</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E44:33_1_1"/><b>then </b><i><font color="Green">E546</font></i>: 
<font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E45:33_1_1"/><b>then </b>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E46:33_1_1"/><b>then </b><i><font color="Green">E547</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as  non <a href="xboole_0.html#V1">empty</a> non <a href="funct_1.html#V3">constant</a> <a href="topreal1.html#V1">special</a> <a href="topreal1.html#V2">unfolded</a> <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, , <a class="ref" href="jordan8.html#T7">JORDAN8:7</a>, <a class="ref" href="jordan8.html#T8">JORDAN8:8</a></i>;<br/>
<b>set </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>;<br/>
<a NAME="E48:33_1_1"/><i><font color="Green">E548</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E49:33_1_1"/>
( <font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 )
 
<b>by </b><i><a class="ref" href="jordan3.html#T7">JORDAN3:7</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E50:33_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E51:33_1_1"/><b>then </b><i><font color="Green">E549</font></i>: 
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E52:33_1_1"/><b>then </b><i><font color="Green">E550</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E53:33_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E55:33_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<b>then </b><i><font color="Green">E551</font></i>: <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font>
<b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>
;<br/>
<a NAME="E57:33_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>
<b>then </b><i><font color="Green">E552</font></i>: <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> = 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E59:33_1_1"/><i><font color="Green">E553</font></i>: 
<font color="Maroon">g</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="jordan8.html#T5">JORDAN8:5</a></i>;<br/>
<a NAME="E60:33_1_1"/><b>then </b><i><font color="Green">E554</font></i>: 
<font color="Maroon">g</font> is <a href="goboard5.html#V2">standard</a>
 
<b>by </b><i><a class="ref" href="jordan8.html#T7">JORDAN8:7</a></i>;<br/>
<a NAME="E61:33_1_1"/><i><font color="Green">E555</font></i>: 
for <font color="Olive">j</font>, <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font> holds <br/><font color="Olive">i</font> <a href="hidden.html#R1">=</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_17"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_17"/><i><font color="Green">E556</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 <b>and </b><br/><a NAME="E2:33_1_1_17"/><i><font color="Green">E557</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 <b>and </b><br/><a NAME="E3:33_1_1_17"/><i><font color="Green">E558</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E4:33_1_1_17"/><i><font color="Green">E559</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E5:33_1_1_17"/>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:33_1_1_17"/><b>then </b><i><font color="Green">E560</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>;<br/>
<a NAME="E7:33_1_1_17"/>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:33_1_1_17"/><b>then </b><i><font color="Green">E561</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_17_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_17_1"/>
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> or <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_17_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_17_1_1"/><i><font color="Green">E562</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">l</font> = <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>;<br/><b>set </b><font color="Maroon">m'</font> = <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>;<br/><a NAME="E2:33_1_1_17_1_1"/><i><font color="Green">E564</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:33_1_1_17_1_1"/><i><font color="Green">E565</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E4:33_1_1_17_1_1"/><b>then </b><i><font color="Green">E566</font></i>: 
<font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E5:33_1_1_17_1_1"/>
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:33_1_1_17_1_1"/><b>then </b><i><font color="Green">E567</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:33_1_1_17_1_1"/><i><font color="Green">E568</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E8:33_1_1_17_1_1"/>
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:33_1_1_17_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:33_1_1_17_1_1"/><b>then </b><i><font color="Green">E569</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:33_1_1_17_1_1"/><b>then </b><i><font color="Green">E570</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><a NAME="E12:33_1_1_17_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:33_1_1_17_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><b>hence </b><a NAME="E14:33_1_1_17_1_1"/>
contradiction
 <b>by </b><i>, , , , , , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_17_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_17_1_2"/><i><font color="Green">E571</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">l</font> = <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>;<br/><b>set </b><font color="Maroon">m'</font> = <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>;<br/><a NAME="E2:33_1_1_17_1_2"/><i><font color="Green">E572</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:33_1_1_17_1_2"/><i><font color="Green">E573</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E4:33_1_1_17_1_2"/><b>then </b><i><font color="Green">E574</font></i>: 
<font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T6" target="_self">Th6</a></i>;<br/><a NAME="E5:33_1_1_17_1_2"/>
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:33_1_1_17_1_2"/><b>then </b><i><font color="Green">E575</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:33_1_1_17_1_2"/><i><font color="Green">E576</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E8:33_1_1_17_1_2"/>
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:33_1_1_17_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:33_1_1_17_1_2"/><b>then </b><i><font color="Green">E577</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:33_1_1_17_1_2"/><b>then </b><i><font color="Green">E578</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><a NAME="E12:33_1_1_17_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:33_1_1_17_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><b>hence </b><a NAME="E14:33_1_1_17_1_2"/>
contradiction
 <b>by </b><i>, , , , , , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E62:33_1_1"/><i><font color="Green">E579</font></i>: 
for <font color="Olive">j</font>, <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_18"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_18"/><i><font color="Green">E580</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E2:33_1_1_18"/><i><font color="Green">E581</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E3:33_1_1_18"/><i><font color="Green">E582</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:33_1_1_18"/><i><font color="Green">E583</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E5:33_1_1_18"/><i><font color="Green">E584</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:33_1_1_18"/><i><font color="Green">E585</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:33_1_1_18"/><b>then </b><i><font color="Green">E586</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_18_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_18_1"/>
( <font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> or <font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_18_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_18_1_1"/>
<font color="Maroon">j</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><a NAME="E2:33_1_1_18_1_1"/><b>then </b>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_18_1_1"/>
contradiction
 <b>by </b><i>, , , , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_18_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_18_1_2"/>
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:33_1_1_18_1_2"/>
contradiction
 <b>by </b><i>, , , , , , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E63:33_1_1"/><i><font color="Green">E587</font></i>: 
for <font color="Olive">j</font>, <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> holds <br/><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_19"/><i><font color="Green">E588</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E2:33_1_1_19"/><i><font color="Green">E589</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E3:33_1_1_19"/><i><font color="Green">E590</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>and </b><br/><a NAME="E4:33_1_1_19"/><i><font color="Green">E591</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 ;<br/>

<a NAME="E5:33_1_1_19"/><i><font color="Green">E592</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:33_1_1_19"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:33_1_1_19"/>
contradiction
 <b>by </b><i>, , , , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E64:33_1_1"/><i><font color="Green">E593</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Olive">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Olive">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_20"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:33_1_1_20"/><i><font color="Green">E594</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/>

<b>consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_20"/><i><font color="Green">E595</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:33_1_1_20"/><i><font color="Green">E596</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font>
 <b>and </b><br/><a NAME="E5:33_1_1_20"/><i><font color="Green">E597</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:33_1_1_20"/><i><font color="Green">E598</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font>
 <b>and </b><br/><a NAME="E7:33_1_1_20"/><i><font color="Green">E599</font></i>: 
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/>
<a NAME="E8:33_1_1_20"/><i><font color="Green">E600</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E9:33_1_1_20"/><i><font color="Green">E601</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E10:33_1_1_20"/><i><font color="Green">E602</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">j2</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_20_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_20_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_20_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_20_1_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_20_1_1"/><b>then </b>
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="gobrd13.html#D2">GOBRD13:def 2</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_20_1_1"/>
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="gobrd11.html#T35">GOBRD11:35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_20_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_20_1_2"/><i><font color="Green">E603</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_20_1_2"/><i><font color="Green">E604</font></i>: 
<font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E3:33_1_1_20_1_2"/>
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , , , , , , , <a class="ref" href="gobrd13.html#D2">GOBRD13:def 2</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_20_1_2"/>
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="gobrd11.html#T35">GOBRD11:35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_20_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_20_1_3"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_20_1_3"/><b>then </b>
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="gobrd13.html#D2">GOBRD13:def 2</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_20_1_3"/>
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="gobrd11.html#T35">GOBRD11:35</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_20_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_20_1_4"/><i><font color="Green">E605</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_20_1_4"/><i><font color="Green">E606</font></i>: 
<font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E3:33_1_1_20_1_4"/>
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="gobrd13.html#D2">GOBRD13:def 2</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_20_1_4"/>
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">i</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="gobrd11.html#T35">GOBRD11:35</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E65:33_1_1"/><i><font color="Green">E607</font></i>: 
<font color="Maroon">g</font> is <a href="goboard5.html#V1">s.c.c.</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_21"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="goboard5.html#D4">GOBOARD5:def 4</a><br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_21"/><i><font color="Green">E608</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E2:33_1_1_21"/><i><font color="Green">E609</font></i>: 
( ( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> ) or <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 ;<br/>

<a NAME="E3:33_1_1_21"/><i><font color="Green">E610</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E4:33_1_1_21"/><b>then </b><i><font color="Green">E611</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:33_1_1_21"/><i><font color="Green">E612</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E6:33_1_1_21"/><i><font color="Green">E613</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E7:33_1_1_21"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:33_1_1_21"/><b>then </b><i><font color="Green">E614</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_21_1"/>
( ( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> ) or ( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> ) or ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> ) )
 </span><b>by </b><i>, <a class="ref" href="nat_1.html#T39">NAT_1:39</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1"/><i><font color="Green">E615</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1"/><b>then </b><i><font color="Green">E616</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:33_1_1_21_1_1"/><b>then </b><i><font color="Green">E617</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E4:33_1_1_21_1_1"/><i><font color="Green">E618</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:33_1_1_21_1_1"/><i><font color="Green">E619</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E7:33_1_1_21_1_1"/><i><font color="Green">E620</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E8:33_1_1_21_1_1"/><i><font color="Green">E621</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font>
 <b>and </b><br/><a NAME="E9:33_1_1_21_1_1"/><i><font color="Green">E622</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E10:33_1_1_21_1_1"/><i><font color="Green">E623</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font>
 <b>and </b><br/><a NAME="E11:33_1_1_21_1_1"/><i><font color="Green">E624</font></i>: 
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E12:33_1_1_21_1_1"/><i><font color="Green">E625</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E13:33_1_1_21_1_1"/><i><font color="Green">E626</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E14:33_1_1_21_1_1"/><i><font color="Green">E627</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E15:33_1_1_21_1_1"/><b>then </b><i><font color="Green">E628</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">i1'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E17:33_1_1_21_1_1"/><i><font color="Green">E629</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E18:33_1_1_21_1_1"/><i><font color="Green">E630</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font>
 <b>and </b><br/><a NAME="E19:33_1_1_21_1_1"/><i><font color="Green">E631</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E20:33_1_1_21_1_1"/><i><font color="Green">E632</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font>
 <b>and </b><br/><a NAME="E21:33_1_1_21_1_1"/><i><font color="Green">E633</font></i>: 
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E22:33_1_1_21_1_1"/><i><font color="Green">E634</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">i1'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">j1'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T12" target="_self">Th12</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E23:33_1_1_21_1_1"/><i><font color="Green">E635</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">i2'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2'</font> &amp; <font color="Maroon">j2'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T14" target="_self">Th14</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><b>assume </b><a NAME="E24:33_1_1_21_1_1"/><i><font color="Green">E636</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_21_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_21_1_1_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T16" target="_self">Th16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_1"/><i><font color="Green">E637</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_1"/><b>then </b><i><font color="Green">E638</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font>
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T21">GOBOARD7:21</a></i>;<br/><a NAME="E3:33_1_1_21_1_1_1_1_1"/>
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> or <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="goboard7.html#T24">GOBOARD7:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_1_1_1_1"/>
contradiction
 <b>by </b><i>, , , , , , , , , , <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan9.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_2"/><i><font color="Green">E639</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_2"/><b>then </b>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_1_1_1_2"/>
contradiction
 <b>by </b><i>, , , , , , , , , <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_3"/><i><font color="Green">E640</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_3"/><b>then </b>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_1_1_1_3"/>
contradiction
 <b>by </b><i>, , , , , , , , , <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_4"/><i><font color="Green">E641</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_4"/><b>then </b><i><font color="Green">E642</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font>
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T21">GOBOARD7:21</a></i>;<br/><a NAME="E3:33_1_1_21_1_1_1_1_4"/>
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> or <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="goboard7.html#T24">GOBOARD7:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_1_1_1_4"/>
contradiction
 <b>by </b><i>, , , , , , , , <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan9.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_5"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_5"/><i><font color="Green">E643</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_5"/><b>then </b>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_1_1_1_5"/>
contradiction
 <b>by </b><i>, , , , , , , , , <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_6"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_6"/><i><font color="Green">E644</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_6"/><b>then </b><i><font color="Green">E645</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font>
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T22">GOBOARD7:22</a></i>;<br/><a NAME="E3:33_1_1_21_1_1_1_1_6"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> or <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="goboard7.html#T25">GOBOARD7:25</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_1_1_1_6"/>
contradiction
 <b>by </b><i>, , , , , , , , , , <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan9.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_7"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_7"/><i><font color="Green">E646</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_7"/><b>then </b><i><font color="Green">E647</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font>
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T22">GOBOARD7:22</a></i>;<br/><a NAME="E3:33_1_1_21_1_1_1_1_7"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> or <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="goboard7.html#T25">GOBOARD7:25</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_1_1_1_7"/>
contradiction
 <b>by </b><i>, , , , , , , , <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan9.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_8"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_8"/><i><font color="Green">E648</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_8"/><b>then </b>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_1_1_1_8"/>
contradiction
 <b>by </b><i>, , , , , , , , , <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_9"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_9"/><i><font color="Green">E649</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_9"/><b>then </b>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_1_1_1_9"/>
contradiction
 <b>by </b><i>, , , , , , , , , <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_10"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_10"/><i><font color="Green">E650</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_10"/><b>then </b><i><font color="Green">E651</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font>
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T22">GOBOARD7:22</a></i>;<br/><a NAME="E3:33_1_1_21_1_1_1_1_10"/>
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> or <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="goboard7.html#T25">GOBOARD7:25</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_1_1_1_10"/>
contradiction
 <b>by </b><i>, , , , , , , , <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan9.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_11"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_11"/><i><font color="Green">E652</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_11"/><b>then </b><i><font color="Green">E653</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font>
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T22">GOBOARD7:22</a></i>;<br/><a NAME="E3:33_1_1_21_1_1_1_1_11"/>
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> or <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="goboard7.html#T25">GOBOARD7:25</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_1_1_1_11"/>
contradiction
 <b>by </b><i>, , , , , , <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan9.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_12"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_12"/><i><font color="Green">E654</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_12"/><b>then </b>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_1_1_1_12"/>
contradiction
 <b>by </b><i>, , , , , , , , , <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_13"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_13"/><i><font color="Green">E655</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_13"/><b>then </b><i><font color="Green">E656</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font>
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T21">GOBOARD7:21</a></i>;<br/><a NAME="E3:33_1_1_21_1_1_1_1_13"/>
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> or <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="goboard7.html#T24">GOBOARD7:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_1_1_1_13"/>
contradiction
 <b>by </b><i>, , , , , , , , <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan9.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_14"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_14"/><i><font color="Green">E657</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_14"/><b>then </b>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_1_1_1_14"/>
contradiction
 <b>by </b><i>, , , , , , , , , <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_15"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_15"/><i><font color="Green">E658</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_15"/><b>then </b>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_1_1_1_15"/>
contradiction
 <b>by </b><i>, , , , , , , , , <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_1_1_1_16"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_1_1_1_16"/><i><font color="Green">E659</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_1_1_1_16"/><b>then </b><i><font color="Green">E660</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font>
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T21">GOBOARD7:21</a></i>;<br/><a NAME="E3:33_1_1_21_1_1_1_1_16"/>
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> or <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 <b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="goboard7.html#T24">GOBOARD7:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_1_1_1_16"/>
contradiction
 <b>by </b><i>, , , , , , <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan9.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E26:33_1_1_21_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_2"/>
( <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_2"/>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T65">XBOOLE_1:65</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3"/><i><font color="Green">E661</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3"/><b>then </b><i><font color="Green">E662</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:33_1_1_21_1_3"/><b>then </b><i><font color="Green">E663</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E4:33_1_1_21_1_3"/><i><font color="Green">E664</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:33_1_1_21_1_3"/><i><font color="Green">E665</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:33_1_1_21_1_3"/><i><font color="Green">E666</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:33_1_1_21_1_3"/><i><font color="Green">E667</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E9:33_1_1_21_1_3"/><i><font color="Green">E668</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font>
 <b>and </b><br/><a NAME="E10:33_1_1_21_1_3"/><i><font color="Green">E669</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E11:33_1_1_21_1_3"/><i><font color="Green">E670</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font>
 <b>and </b><br/><a NAME="E12:33_1_1_21_1_3"/><i><font color="Green">E671</font></i>: 
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E13:33_1_1_21_1_3"/><i><font color="Green">E672</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">i1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1</font> &amp; <font color="Maroon">j1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E14:33_1_1_21_1_3"/><i><font color="Green">E673</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E15:33_1_1_21_1_3"/><i><font color="Green">E674</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">i1'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E17:33_1_1_21_1_3"/><i><font color="Green">E675</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1'</font>,<font color="Maroon">j1'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E18:33_1_1_21_1_3"/><i><font color="Green">E676</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1'</font>,<font color="Maroon">j1'</font>
 <b>and </b><br/><a NAME="E19:33_1_1_21_1_3"/><i><font color="Green">E677</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2'</font>,<font color="Maroon">j2'</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E20:33_1_1_21_1_3"/><i><font color="Green">E678</font></i>: 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2'</font>,<font color="Maroon">j2'</font>
 <b>and </b><br/><a NAME="E21:33_1_1_21_1_3"/><i><font color="Green">E679</font></i>: 
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E22:33_1_1_21_1_3"/><i><font color="Green">E680</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">i1'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j1'</font> &amp; <font color="Maroon">j1'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T12" target="_self">Th12</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E23:33_1_1_21_1_3"/><i><font color="Green">E681</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">i2'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2'</font> &amp; <font color="Maroon">j2'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T14" target="_self">Th14</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><b>assume </b><a NAME="E24:33_1_1_21_1_3"/><i><font color="Green">E682</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">i</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">g</font>,<font color="Maroon">j</font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_21_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_21_1_3_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i>, <a class="ref" href="jordan9.html#T16" target="_self">Th16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_1"/><i><font color="Green">E683</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_1"/><b>then </b><i><font color="Green">E684</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T21">GOBOARD7:21</a></i>;<br/><a NAME="E3:33_1_1_21_1_3_1_1_1"/>
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> or <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="goboard7.html#T24">GOBOARD7:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_3_1_1_1"/>
contradiction
 <b>by </b><i>, , , , , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan9.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_2"/><i><font color="Green">E685</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_2"/><b>then </b>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_3_1_1_2"/>
contradiction
 <b>by </b><i>, , , , , , , , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_3"/><i><font color="Green">E686</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_3"/><b>then </b>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_3_1_1_3"/>
contradiction
 <b>by </b><i>, , , , , , , , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_4"/><i><font color="Green">E687</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_4"/><b>then </b><i><font color="Green">E688</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T21">GOBOARD7:21</a></i>;<br/><a NAME="E3:33_1_1_21_1_3_1_1_4"/>
( <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> or <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="goboard7.html#T24">GOBOARD7:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_3_1_1_4"/>
contradiction
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan9.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_5"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_5"/><i><font color="Green">E689</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_5"/><b>then </b>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_3_1_1_5"/>
contradiction
 <b>by </b><i>, , , , , , , , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_6"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_6"/><i><font color="Green">E690</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_6"/><b>then </b><i><font color="Green">E691</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T22">GOBOARD7:22</a></i>;<br/><a NAME="E3:33_1_1_21_1_3_1_1_6"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> or <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="goboard7.html#T25">GOBOARD7:25</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_3_1_1_6"/>
contradiction
 <b>by </b><i>, , , , , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan9.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_7"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_7"/><i><font color="Green">E692</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_7"/><b>then </b><i><font color="Green">E693</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T22">GOBOARD7:22</a></i>;<br/><a NAME="E3:33_1_1_21_1_3_1_1_7"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> or <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="goboard7.html#T25">GOBOARD7:25</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_3_1_1_7"/>
contradiction
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan9.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_8"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_8"/><i><font color="Green">E694</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_8"/><b>then </b>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> &amp; <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_3_1_1_8"/>
contradiction
 <b>by </b><i>, , , , , , , , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_9"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_9"/><i><font color="Green">E695</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_9"/><b>then </b>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_3_1_1_9"/>
contradiction
 <b>by </b><i>, , , , , , , , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_10"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_10"/><i><font color="Green">E696</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_10"/><b>then </b><i><font color="Green">E697</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T22">GOBOARD7:22</a></i>;<br/><a NAME="E3:33_1_1_21_1_3_1_1_10"/>
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> or <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="goboard7.html#T25">GOBOARD7:25</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_3_1_1_10"/>
contradiction
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan9.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_11"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_11"/><i><font color="Green">E698</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_11"/><b>then </b><i><font color="Green">E699</font></i>: 
<font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T22">GOBOARD7:22</a></i>;<br/><a NAME="E3:33_1_1_21_1_3_1_1_11"/>
( <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> or <font color="Maroon">i2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="goboard7.html#T25">GOBOARD7:25</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_3_1_1_11"/>
contradiction
 <b>by </b><i>, , , , , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan9.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_12"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_12"/><i><font color="Green">E700</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_12"/><b>then </b>
( ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) or ( <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> ) )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_3_1_1_12"/>
contradiction
 <b>by </b><i>, , , , , , , , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_13"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_13"/><i><font color="Green">E701</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_13"/><b>then </b><i><font color="Green">E702</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T21">GOBOARD7:21</a></i>;<br/><a NAME="E3:33_1_1_21_1_3_1_1_13"/>
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> or <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="goboard7.html#T24">GOBOARD7:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_3_1_1_13"/>
contradiction
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan9.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_14"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_14"/><i><font color="Green">E703</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_14"/><b>then </b>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_3_1_1_14"/>
contradiction
 <b>by </b><i>, , , , , , , , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_15"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_15"/><i><font color="Green">E704</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_15"/><b>then </b>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1'</font> ) )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_21_1_3_1_1_15"/>
contradiction
 <b>by </b><i>, , , , , , , , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_21_1_3_1_1_16"><b>suppose </b></a><a NAME="E1:33_1_1_21_1_3_1_1_16"/><i><font color="Green">E705</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">i1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2'</font> &amp; <font color="Maroon">j1'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_21_1_3_1_1_16"/><b>then </b><i><font color="Green">E706</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i1'</font>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="goboard7.html#T21">GOBOARD7:21</a></i>;<br/><a NAME="E3:33_1_1_21_1_3_1_1_16"/>
( <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> or <font color="Maroon">j2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2'</font> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="goboard7.html#T24">GOBOARD7:24</a></i>;<br/><b>hence </b><a NAME="E4:33_1_1_21_1_3_1_1_16"/>
contradiction
 <b>by </b><i>, , , , , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T17" target="_self">Th17</a>, <a class="ref" href="jordan9.html#T18" target="_self">Th18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E26:33_1_1_21_1_3"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E66:33_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i>, , <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E67:33_1_1"/><b>then </b><i><font color="Green">E707</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E68:33_1_1"/><b>then </b><i><font color="Green">E708</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K3">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
;<br/>
<a NAME="E69:33_1_1"/>
not <font color="Maroon">g</font> is <a href="funct_1.html#V3">constant</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_22"><b>proof </b></a><div class="add">

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

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

<b>thus </b><a NAME="E1:33_1_1_22"/><i><font color="Green">E709</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>

<b>thus </b><a NAME="E2:33_1_1_22"/><i><font color="Green">E710</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>

<a NAME="E3:33_1_1_22"/><b>then </b>
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, , <a class="ref" href="goboard7.html#T31">GOBOARD7:31</a></i>;<br/>
<a NAME="E4:33_1_1_22"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:33_1_1_22"/>
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 2
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g</font> as  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a> <b>by </b><i>, , , , , <a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a>, <a class="ref" href="jordan8.html#T7">JORDAN8:7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_23"/>
<font color="Maroon">m</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/><a NAME="E2:33_1_1_23"/><b>then </b><i><font color="Green">E711</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:33_1_1_23"/><i><font color="Green">E712</font></i>: 
(  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">g</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp;  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">g</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K3">`</a>  )
 <b>by </b><i><a class="ref" href="goboard9.html#D1">GOBOARD9:def 1</a>, <a class="ref" href="goboard9.html#D2">GOBOARD9:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">Lg'</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K3">`</a>  as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><a NAME="E5:33_1_1_23"/><i><font color="Green">E714</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Lg'</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_1"><b>proof </b></a><div class="add">

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

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_23_1"/><i><font color="Green">E716</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:33_1_1_23_1"/><i><font color="Green">E717</font></i>: 
not <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Lg'</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c</font> = <font color="Maroon">c</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<a NAME="E4:33_1_1_23_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/>
<b>then consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:33_1_1_23_1"/><i><font color="Green">E718</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font> )
 <b>and </b><br/><a NAME="E7:33_1_1_23_1"/><i><font color="Green">E719</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="sppol_2.html#T14">SPPOL_2:14</a></i>;<br/>
<a NAME="E8:33_1_1_23_1"/>
( <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font> )
 
<b>by </b><i>, <a class="ref" href="goboard2.html#T3">GOBOARD2:3</a></i>;<br/>
<a NAME="E9:33_1_1_23_1"/><b>then </b><i><font color="Green">E720</font></i>: 
( <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>;<br/>
<a NAME="E10:33_1_1_23_1"/><i><font color="Green">E721</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E11:33_1_1_23_1"/>
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E12:33_1_1_23_1"/><b>then </b><i><font color="Green">E722</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E13:33_1_1_23_1"/><b>then </b><i><font color="Green">E723</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i/>;<br/>
<a NAME="E14:33_1_1_23_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i>, , , , , <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E15:33_1_1_23_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="gobrd13.html#T30">GOBRD13:30</a></i>;<br/>
<a NAME="E16:33_1_1_23_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E17:33_1_1_23_1"/><b>then </b>
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E18:33_1_1_23_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div><a NAME="E6:33_1_1_23"/><i><font color="Green">E724</font></i>: 
 <a href="euclid.html#K15">TOP-REAL</a> 2 <a href="hidden.html#R1">=</a>  <a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="euclid.html#D8">EUCLID:def 8</a></i>;<br/><a NAME="E7:33_1_1_23"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="sppol_1.html#T49">SPPOL_1:49</a></i>;<br/><a NAME="E8:33_1_1_23"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/><a NAME="E9:33_1_1_23"/><b>then </b><i><font color="Green">E725</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K3">`</a>  <a href="hidden.html#R1">=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/><a NAME="E10:33_1_1_23"/><i><font color="Green">E726</font></i>: 
for <font color="Olive">j</font>, <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Olive">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_23_2"/><i><font color="Green">E727</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E2:33_1_1_23_2"/><i><font color="Green">E728</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/>

<a NAME="E3:33_1_1_23_2"/><i><font color="Green">E729</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E4:33_1_1_23_2"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, </i>;<br/>
<a NAME="E5:33_1_1_23_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E6:33_1_1_23_2"/><b>then </b>
<font color="Maroon">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E7:33_1_1_23_2"/>
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:33_1_1_23"/><i><font color="Green">E730</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan3.html#T75">JORDAN3:75</a></i>;<br/><a NAME="E12:33_1_1_23"/><i><font color="Green">E731</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 holds <br/> not <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:33_1_1_23_3"/><i><font color="Green">E732</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 <b>and </b><br/><a NAME="E2:33_1_1_23_3"/><i><font color="Green">E733</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1
 ;<br/>

<a NAME="E3:33_1_1_23_3"/><i><font color="Green">E734</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>set </b><font color="Maroon">p</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>;<br/>
<b>assume </b><a NAME="E4:33_1_1_23_3"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 ;<br/>

<b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:33_1_1_23_3"/><i><font color="Green">E735</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E7:33_1_1_23_3"/><i><font color="Green">E736</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E8:33_1_1_23_3"/><i><font color="Green">E737</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>
 <b>by </b><i>, </i>;<br/>
<a NAME="E9:33_1_1_23_3"/><i><font color="Green">E738</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:33_1_1_23_3"/><b>then </b><i><font color="Green">E739</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 
<b>by </b><i><a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:33_1_1_23_3"/><i><font color="Green">E740</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E12:33_1_1_23_3"/><b>then </b><i><font color="Green">E741</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 
<b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:33_1_1_23_3"/><i><font color="Green">E742</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E14:33_1_1_23_3"/><i><font color="Green">E743</font></i>: 
( 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E15:33_1_1_23_3"/><i><font color="Green">E744</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K2">Values</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, </i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_23_3_1"/>
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font> or <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span><b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , <a class="ref" href="jordan9.html#T12" target="_self">Th12</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_3_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_23_3_1_1"/><i><font color="Green">E745</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_3_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, </i>;<br/><a NAME="E3:33_1_1_23_3_1_1"/><b>then </b><i><font color="Green">E746</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:33_1_1_23_3_1_1"/><i><font color="Green">E747</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
<b>by </b><i>, , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font>
<b>by </b><i>, , , <a class="ref" href="jordan9.html#T14" target="_self">Th14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">j</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>
;<br/><b>hence </b><a NAME="E6:33_1_1_23_3_1_1"/>
contradiction
 <b>by </b><i>, , , <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_3_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_23_3_1_2"/><i><font color="Green">E748</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_3_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_3_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_23_3_1_2_1_1"/>
( <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> or <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font> )
 </span><b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_3_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_23_3_1_2_1_1_1"/><i><font color="Green">E749</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_3_1_2_1_1_1"/>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, </i>;<br/><a NAME="E3:33_1_1_23_3_1_2_1_1_1"/><b>then </b><i><font color="Green">E750</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:33_1_1_23_3_1_2_1_1_1"/><i><font color="Green">E751</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, </i>;<br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
<b>by </b><i>, , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font>
<b>by </b><i>, , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>
;<br/><b>hence </b><a NAME="E6:33_1_1_23_3_1_2_1_1_1"/>
contradiction
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T16" target="_self">Th16</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_3_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_23_3_1_2_1_1_2"/><i><font color="Green">E752</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">l</font> = <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1;<br/><a NAME="E2:33_1_1_23_3_1_2_1_1_2"/><i><font color="Green">E753</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E3:33_1_1_23_3_1_2_1_1_2"/><i><font color="Green">E754</font></i>: 
( <font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a>, <a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E4:33_1_1_23_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E755</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E5:33_1_1_23_3_1_2_1_1_2"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E6:33_1_1_23_3_1_2_1_1_2"/><b>then </b><i><font color="Green">E756</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:33_1_1_23_3_1_2_1_1_2"/><i><font color="Green">E757</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, </i>;<br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font>
<b>by </b><i>, , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, <a class="ref" href="jordan9.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T16" target="_self">Th16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>
;<br/><b>hence </b><a NAME="E9:33_1_1_23_3_1_2_1_1_2"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, <a class="ref" href="jordan9.html#T16" target="_self">Th16</a>, , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:33_1_1_23_3_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E13:33_1_1_23"/><i><font color="Green">E758</font></i>: 
for <font color="Olive">h</font> being non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>  st  <a href="topreal1.html#K5">L~</a> <font color="Olive">h</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font> holds <br/>for <font color="Olive">Comp</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">Comp</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Olive">h</font></span>)</span> <a href="subset_1.html#K3">`</a>  holds <br/>for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">n</font> &amp; <font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a> <font color="Olive">Comp</font> &amp; not <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">n</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Olive">h</font> holds <br/><font color="Maroon">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">Comp</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">h</font> be  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/>

<b>assume </b><a NAME="E1:33_1_1_23_4"/><i><font color="Green">E761</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 ;<br/>

<b>let </b><font color="Maroon">Comp</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>assume </b><a NAME="E2:33_1_1_23_4"/><i><font color="Green">E762</font></i>: 
<font color="Maroon">Comp</font> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">h</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:33_1_1_23_4"/><i><font color="Green">E763</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 <b>and </b><br/><a NAME="E4:33_1_1_23_4"/><i><font color="Green">E764</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Comp</font>
 <b>and </b><br/><a NAME="E5:33_1_1_23_4"/><i><font color="Green">E765</font></i>: 
not <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">h</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">h</font></span>)</span> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E7:33_1_1_23_4"/><i><font color="Green">E766</font></i>: 
<font color="Maroon">rc</font> <a href="tarski.html#R1">c=</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E8:33_1_1_23_4"/><i><font color="Green">E767</font></i>: 
<font color="Maroon">rc</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">h</font></span>)</span> <a href="subset_1.html#K3">`</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="subset_1.html#T32">SUBSET_1:32</a></i>;<br/>
<a NAME="E9:33_1_1_23_4"/><i><font color="Green">E768</font></i>: 
<font color="Maroon">rc</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_4_1"><b>proof </b></a><div class="add">

<a NAME="E1:33_1_1_23_4_1"/>
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_1_1_23_4_1"/><i><font color="Green">E769</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_4_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">a</font> = <font color="Maroon">p</font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_4_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_23_4_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">h</font>
 ;<br/><b>then consider </b><font color="Maroon">j</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_23_4_1_1_1"/><i><font color="Green">E770</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E4:33_1_1_23_4_1_1_1"/><i><font color="Green">E771</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 <b>and </b><br/><a NAME="E5:33_1_1_23_4_1_1_1"/><i><font color="Green">E772</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>
 <b>by </b><i>, <a class="ref" href="sppol_2.html#T13">SPPOL_2:13</a></i>;<br/><a NAME="E6:33_1_1_23_4_1_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T12" target="_self">Th12</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="gobrd13.html#T30">GOBRD13:30</a></i>;<br/><a NAME="E7:33_1_1_23_4_1_1_1"/><b>then </b><i><font color="Green">E773</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:33_1_1_23_4_1_1_1"/>
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">j</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E9:33_1_1_23_4_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="jordan9.html#T14" target="_self">Th14</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:33_1_1_23_4_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rc</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>thus </b><a NAME="E3:33_1_1_23_4_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:33_1_1_23_4_1"/>
<font color="Maroon">rc</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:33_1_1_23_4"/><i><font color="Green">E774</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, </i>;<br/>
<a NAME="E11:33_1_1_23_4"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, </i>;<br/>
<a NAME="E12:33_1_1_23_4"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">h</font>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E13:33_1_1_23_4"/><b>then </b><i><font color="Green">E775</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">h</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E14:33_1_1_23_4"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E15:33_1_1_23_4"/><b>then </b><i><font color="Green">E776</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">rc</font>
 
<b>by </b><i>, , <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E16:33_1_1_23_4"/>
<font color="Maroon">rc</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, </i>;<br/>
<a NAME="E17:33_1_1_23_4"/><b>then </b><i><font color="Green">E777</font></i>: 
<font color="Maroon">rc</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, <a class="ref" href="jordan9.html#T12" target="_self">Th12</a>, <a class="ref" href="connsp_1.html#T19">CONNSP_1:19</a></i>;<br/>
<a NAME="E18:33_1_1_23_4"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">n</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, </i>;<br/>
<a NAME="E19:33_1_1_23_4"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">rc</font>
 
<b>by </b><i><a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E20:33_1_1_23_4"/><b>then </b><i><font color="Green">E778</font></i>: 
<font color="Maroon">rc</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Comp</font>
 
<b>by </b><i><a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E21:33_1_1_23_4"/>
<font color="Maroon">rc</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">h</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E22:33_1_1_23_4"/><b>then </b>
<font color="Maroon">rc</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Comp</font>
 
<b>by </b><i><a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T14" target="_self">Th14</a>, <a class="ref" href="goboard9.html#T6">GOBOARD9:6</a></i>;<br/>
<b>hence </b><a NAME="E23:33_1_1_23_4"/>
<font color="Maroon">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">Comp</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div><a NAME="E14:33_1_1_23"/><i><font color="Green">E779</font></i>: 
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_5"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">rg</font> =  <a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font>;<br/>
<a NAME="E1:33_1_1_23_5"/><i><font color="Green">E781</font></i>: 
 <a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:33_1_1_23_5"/><i><font color="Green">E782</font></i>: 
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/>
<a NAME="E3:33_1_1_23_5"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/>
<a NAME="E4:33_1_1_23_5"/><b>then </b><i><font color="Green">E783</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>
<a NAME="E5:33_1_1_23_5"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:33_1_1_23_5"/><b>then </b><i><font color="Green">E784</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<b>set </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1;<br/>
<b>set </b><font color="Maroon">q</font> = <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2;<br/>
<b>set </b><font color="Maroon">a</font> = <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">l</font> = <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>;<br/>
<a NAME="E7:33_1_1_23_5"/><i><font color="Green">E785</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_5.html#T68">FINSEQ_5:68</a></i>;<br/>
<a NAME="E8:33_1_1_23_5"/><i><font color="Green">E786</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E9:33_1_1_23_5"/><b>then </b><i><font color="Green">E787</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:33_1_1_23_5"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/>
<a NAME="E11:33_1_1_23_5"/><b>then </b><i><font color="Green">E788</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>
<a NAME="E12:33_1_1_23_5"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E13:33_1_1_23_5"/><b>then </b><i><font color="Green">E789</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 
;<br/>
<a NAME="E14:33_1_1_23_5"/><i><font color="Green">E790</font></i>: 
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#T56">BINARITH:56</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#T55">BINARITH:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
<b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="real_1.html#K5">-</a> 1</span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">m</font></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i>, , <a class="ref" href="jordan9.html#T12" target="_self">Th12</a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">k</font> <a href="real_1.html#K5">-</a> 1

;<br/>
<a NAME="E16:33_1_1_23_5"/><b>then </b><i><font color="Green">E791</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E17:33_1_1_23_5"/><b>then </b><i><font color="Green">E792</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E18:33_1_1_23_5"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E19:33_1_1_23_5"/><b>then </b><i><font color="Green">E793</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:33_1_1_23_5"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/>
<a NAME="E21:33_1_1_23_5"/><b>then </b><i><font color="Green">E794</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E22:33_1_1_23_5"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font>
 
<b>by </b><i>, <a class="ref" href="sppol_1.html#T5">SPPOL_1:5</a></i>;<br/>
<a NAME="E23:33_1_1_23_5"/><b>then </b><i><font color="Green">E795</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E796</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 = 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan9.html#T16" target="_self">Th16</a>, <a class="ref" href="finseq_5.html#T69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="jordan9.html#T16" target="_self">Th16</a>, <a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>
;<br/>
<a NAME="E25:33_1_1_23_5"/><i><font color="Green">E797</font></i>: 
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , , </i>;<br/>
<a NAME="E26:33_1_1_23_5"/><i><font color="Green">E798</font></i>: 
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a></i>;<br/>
<a NAME="E27:33_1_1_23_5"/>
not <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E28:33_1_1_23_5"/><b>then </b><i><font color="Green">E799</font></i>: 
not <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T22">SPPOL_2:22</a></i>;<br/>
<a NAME="E29:33_1_1_23_5"/><i><font color="Green">E800</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i>, </i>;<br/>
<a NAME="E30:33_1_1_23_5"/><i><font color="Green">E801</font></i>: 
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E31:33_1_1_23_5"/><b>then </b>
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<a NAME="E32:33_1_1_23_5"/><b>then </b><i><font color="Green">E802</font></i>: 
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E33:33_1_1_23_5"/>
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E34:33_1_1_23_5"/><b>then </b>
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E35:33_1_1_23_5"/><b>then </b><i><font color="Green">E803</font></i>: 
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">p1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">p2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">q1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">q2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E37:33_1_1_23_5"/><i><font color="Green">E808</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E38:33_1_1_23_5"/><i><font color="Green">E809</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>and </b><br/><a NAME="E39:33_1_1_23_5"/><i><font color="Green">E810</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">q1</font>,<font color="Maroon">q2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E40:33_1_1_23_5"/><i><font color="Green">E811</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>
 <b>and </b><br/><a NAME="E41:33_1_1_23_5"/><i><font color="Green">E812</font></i>: 
( ( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> ) or ( <font color="Maroon">p1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> ) or ( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> ) or ( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/>
<a NAME="E42:33_1_1_23_5"/><i><font color="Green">E813</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">p1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">p2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_23_5_3"/>
( ( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> ) or ( <font color="Maroon">p1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> ) or ( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> ) or ( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="ref" href="jordan9.html#T24" target="_self">Th24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_1"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_1"/><i><font color="Green">E814</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">a1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">a2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">p'1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">p'2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_23_5_3_1"/><i><font color="Green">E819</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:33_1_1_23_5_3_1"/><i><font color="Green">E820</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>
 <b>and </b><br/><a NAME="E5:33_1_1_23_5_3_1"/><i><font color="Green">E821</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p'1</font>,<font color="Maroon">p'2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:33_1_1_23_5_3_1"/><i><font color="Green">E822</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">p'1</font>,<font color="Maroon">p'2</font>
 <b>and </b><br/><a NAME="E7:33_1_1_23_5_3_1"/><i><font color="Green">E823</font></i>: 
( ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E8:33_1_1_23_5_3_1"/><i><font color="Green">E824</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><b>thus </b><a NAME="E9:33_1_1_23_5_3_1"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_5_3_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_23_5_3_1_1_1"/>
( ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_1_1_1_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_1_1_1_1"/><b>then </b><i><font color="Green">E825</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green">E826</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E4:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green">E827</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, </i>;<br/><a NAME="E5:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green">E828</font></i>: 
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green">E829</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , </i>;<br/><a NAME="E7:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green">E830</font></i>: 
( <font color="Maroon">p2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a2</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><i><font color="Green">E831</font></i>: <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , </i>
;<br/><a NAME="E9:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green">E832</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , </i>;<br/> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> = 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">p2</font>
<b>by </b><i>, , , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , , , <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>
<br/>.= 
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, , <a class="ref" href="jordan9.html#T31" target="_self">Th31</a>, , , <a class="ref" href="gobrd13.html#T35">GOBRD13:35</a></i>
;<br/><a NAME="E11:33_1_1_23_5_3_1_1_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R2">turns_left</a> <font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E12:33_1_1_23_5_3_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="gobrd13.html#R2">turns_left</a> <font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T19" target="_self">Th19</a>, , <a class="ref" href="gobrd13.html#T45">GOBRD13:45</a></i>;<br/><a NAME="E13:33_1_1_23_5_3_1_1_1_1"/><b>then </b><i><font color="Green">E833</font></i>: 
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">p2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , <a class="ref" href="jordan9.html#T19" target="_self">Th19</a>, <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan9.html#T32" target="_self">Th32</a>, <a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/><a NAME="E14:33_1_1_23_5_3_1_1_1_1"/><b>then </b><i><font color="Green">E834</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p1</font> <a href="binarith.html#K3">-'</a> 1 &amp; <font color="Maroon">p1</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><b>set </b><font color="Maroon">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span>;<br/><a NAME="E15:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green">E835</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, </i>;<br/><a NAME="E16:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green">E836</font></i>: 
( <font color="Maroon">p2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> )
 <b>by </b><i>, <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E17:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green">E837</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:33_1_1_23_5_3_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green">E838</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i>, <a class="ref" href="finseq_5.html#T69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan9.html#T19" target="_self">Th19</a>, , <a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>
;<br/><span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, , , <a class="ref" href="finseq_5.html#T68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>
;<br/><a NAME="E21:33_1_1_23_5_3_1_1_1_1"/><b>then </b>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">a2</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , , , <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/><a NAME="E22:33_1_1_23_5_3_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, , <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, , , </i>;<br/><a NAME="E23:33_1_1_23_5_3_1_1_1_1"/><b>then </b><i><font color="Green">E839</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E24:33_1_1_23_5_3_1_1_1_1"/><i><font color="Green">E840</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="sppol_2.html#T22">SPPOL_2:22</a></i>;<br/><a NAME="E25:33_1_1_23_5_3_1_1_1_1"/>
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D2">GOBOARD9:def 2</a></i>;<br/><b>hence </b><a NAME="E26:33_1_1_23_5_3_1_1_1_1"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T18" target="_self">Th18</a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_1_1_1_2"/><i><font color="Green">E841</font></i>: 
( <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_1_1_1_2"/><b>then </b><i><font color="Green">E842</font></i>: 
( <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_1_1_1_2"/><b>then </b>
<font color="Maroon">q1</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><b>then </b> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> = 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">a1</font>,<font color="Maroon">a2</font>
<b>by </b><i>, , , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , , , , <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>
<br/>.= 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>
;<br/><b>hence </b><a NAME="E5:33_1_1_23_5_3_1_1_1_2"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_1_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_1_1_1_3"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_1_1_1_3"/><b>then </b>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><b>then </b> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> = 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
<b>by </b><i>, , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>
<br/>.= 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , , , <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>
;<br/><b>hence </b><a NAME="E4:33_1_1_23_5_3_1_1_1_3"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_1_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_1_1_1_4"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_1_1_1_4"/><b>then </b><i><font color="Green">E843</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i>, , , , , <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan9.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="jordan9.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:33_1_1_23_5_3_1_1_1_4"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , <a class="ref" href="jordan9.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_2"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_2"/><i><font color="Green">E844</font></i>: 
( <font color="Maroon">p1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">a1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">a2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">p'1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">p'2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_23_5_3_2"/><i><font color="Green">E845</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:33_1_1_23_5_3_2"/><i><font color="Green">E846</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>
 <b>and </b><br/><a NAME="E5:33_1_1_23_5_3_2"/><i><font color="Green">E847</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p'1</font>,<font color="Maroon">p'2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:33_1_1_23_5_3_2"/><i><font color="Green">E848</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">p'1</font>,<font color="Maroon">p'2</font>
 <b>and </b><br/><a NAME="E7:33_1_1_23_5_3_2"/><i><font color="Green">E849</font></i>: 
( ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E8:33_1_1_23_5_3_2"/><i><font color="Green">E850</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><b>thus </b><a NAME="E9:33_1_1_23_5_3_2"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_5_3_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_23_5_3_2_1_1"/>
( ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_2_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_2_1_1_1"/><i><font color="Green">E851</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_2_1_1_1"/><b>then </b><i><font color="Green">E852</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_2_1_1_1"/><b>then </b><i><font color="Green">E853</font></i>: 
<font color="Maroon">q2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> = 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">a1</font>,<font color="Maroon">a2</font>
<b>by </b><i>, , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>
<br/>.= 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , , , , <a class="ref" href="jordan9.html#T31" target="_self">Th31</a>, <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>
;<br/><b>hence </b><a NAME="E5:33_1_1_23_5_3_2_1_1_1"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_2_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_2_1_1_2"/>
( <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_2_1_1_2"/><b>then </b><i><font color="Green">E854</font></i>: 
( <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green">E855</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E4:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green">E856</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, </i>;<br/><a NAME="E5:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green">E857</font></i>: 
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green">E858</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , </i>;<br/><a NAME="E7:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green">E859</font></i>: 
( <font color="Maroon">a1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">p1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p1</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><i><font color="Green">E860</font></i>: <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , </i>
;<br/><a NAME="E9:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green">E861</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , </i>;<br/> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> = 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
<b>by </b><i>, , , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , , , <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>
<br/>.= 
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, , <a class="ref" href="jordan9.html#T31" target="_self">Th31</a>, , , <a class="ref" href="gobrd13.html#T37">GOBRD13:37</a></i>
;<br/><a NAME="E11:33_1_1_23_5_3_2_1_1_2"/><b>then </b>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R2">turns_left</a> <font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E12:33_1_1_23_5_3_2_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="gobrd13.html#R2">turns_left</a> <font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T19" target="_self">Th19</a>, , <a class="ref" href="gobrd13.html#T45">GOBRD13:45</a></i>;<br/><a NAME="E13:33_1_1_23_5_3_2_1_1_2"/><b>then </b><i><font color="Green">E862</font></i>: 
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">p1</font>,<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , <a class="ref" href="jordan9.html#T19" target="_self">Th19</a>, <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, <a class="ref" href="jordan9.html#T32" target="_self">Th32</a>, <a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/><a NAME="E14:33_1_1_23_5_3_2_1_1_2"/><b>then </b><i><font color="Green">E863</font></i>: 
<font color="Maroon">p2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><b>set </b><font color="Maroon">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span>;<br/><a NAME="E15:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green">E864</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, </i>;<br/><a NAME="E16:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green">E865</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E17:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green">E866</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:33_1_1_23_5_3_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green">E867</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i>, <a class="ref" href="finseq_5.html#T69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan9.html#T19" target="_self">Th19</a>, , <a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>
;<br/><span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, , , <a class="ref" href="finseq_5.html#T68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>
;<br/><a NAME="E21:33_1_1_23_5_3_2_1_1_2"/><b>then </b>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">a2</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, , , <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/><a NAME="E22:33_1_1_23_5_3_2_1_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, , <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, , , </i>;<br/><a NAME="E23:33_1_1_23_5_3_2_1_1_2"/><b>then </b><i><font color="Green">E868</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E24:33_1_1_23_5_3_2_1_1_2"/><i><font color="Green">E869</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="sppol_2.html#T22">SPPOL_2:22</a></i>;<br/><a NAME="E25:33_1_1_23_5_3_2_1_1_2"/>
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D2">GOBOARD9:def 2</a></i>;<br/><b>hence </b><a NAME="E26:33_1_1_23_5_3_2_1_1_2"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T18" target="_self">Th18</a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_2_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_2_1_1_3"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_2_1_1_3"/><b>then </b><i><font color="Green">E870</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i>, , , , , <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan9.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="jordan9.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:33_1_1_23_5_3_2_1_1_3"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , <a class="ref" href="jordan9.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_2_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_2_1_1_4"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_2_1_1_4"/><b>then </b><i><font color="Green">E871</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> = 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
<b>by </b><i>, , , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , , , <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>
<br/>.= 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>
;<br/><b>hence </b><a NAME="E4:33_1_1_23_5_3_2_1_1_4"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_3"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_3"/><i><font color="Green">E872</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">a1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">a2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">p'1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">p'2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_23_5_3_3"/><i><font color="Green">E873</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:33_1_1_23_5_3_3"/><i><font color="Green">E874</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>
 <b>and </b><br/><a NAME="E5:33_1_1_23_5_3_3"/><i><font color="Green">E875</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p'1</font>,<font color="Maroon">p'2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:33_1_1_23_5_3_3"/><i><font color="Green">E876</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">p'1</font>,<font color="Maroon">p'2</font>
 <b>and </b><br/><a NAME="E7:33_1_1_23_5_3_3"/><i><font color="Green">E877</font></i>: 
( ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E8:33_1_1_23_5_3_3"/><i><font color="Green">E878</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><b>thus </b><a NAME="E9:33_1_1_23_5_3_3"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_5_3_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_23_5_3_3_1_1"/>
( ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_3_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_3_1_1_1"/><i><font color="Green">E879</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_3_1_1_1"/><b>then </b><i><font color="Green">E880</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_3_1_1_1"/><b>then </b><i><font color="Green">E881</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E4:33_1_1_23_5_3_3_1_1_1"/><i><font color="Green">E882</font></i>: 
<font color="Maroon">q2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">a2</font>
 <b>by </b><i>, , <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> = 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">q1</font>,<span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , , , <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>
<br/>.= 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, <a class="ref" href="jordan9.html#T31" target="_self">Th31</a>, , <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>
;<br/><b>hence </b><a NAME="E6:33_1_1_23_5_3_3_1_1_1"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_3_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_3_1_1_2"/>
( <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_3_1_1_2"/><b>then </b>
( <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_3_1_1_2"/><b>then </b><i><font color="Green">E883</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> )
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i>, , , , , <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan9.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="jordan9.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:33_1_1_23_5_3_3_1_1_2"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , <a class="ref" href="jordan9.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_3_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_3_1_1_3"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_3_1_1_3"/><b>then </b><i><font color="Green">E884</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green">E885</font></i>: 
<font color="Maroon">p1</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E4:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green">E886</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E5:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green">E887</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, </i>;<br/><a NAME="E6:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green">E888</font></i>: 
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green">E889</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , </i>;<br/><a NAME="E8:33_1_1_23_5_3_3_1_1_3"/>
<font color="Maroon">p1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p1</font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E9:33_1_1_23_5_3_3_1_1_3"/><b>then </b><i><font color="Green">E890</font></i>: 
( <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p2</font> &amp; <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">p2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><i><font color="Green">E891</font></i>: <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , </i>
;<br/><a NAME="E11:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green">E892</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , </i>;<br/> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> = 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">q1</font>,<span class="p1">(<span class="default"><font color="Maroon">q2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , , , <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>
<br/>.= 
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , <a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, , <a class="ref" href="jordan9.html#T31" target="_self">Th31</a>, , , , <a class="ref" href="gobrd13.html#T39">GOBRD13:39</a></i>
;<br/><a NAME="E13:33_1_1_23_5_3_3_1_1_3"/><b>then </b>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R2">turns_left</a> <font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E14:33_1_1_23_5_3_3_1_1_3"/><b>then </b>
<font color="Maroon">f</font> <a href="gobrd13.html#R2">turns_left</a> <font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T19" target="_self">Th19</a>, <a class="ref" href="jordan9.html#T32" target="_self">Th32</a>, <a class="ref" href="gobrd13.html#T45">GOBRD13:45</a></i>;<br/><a NAME="E15:33_1_1_23_5_3_3_1_1_3"/><b>then </b><i><font color="Green">E893</font></i>: 
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">p1</font>,<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p1</font>,<span class="p2">(<span class="default"><font color="Maroon">p2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , <a class="ref" href="jordan9.html#T19" target="_self">Th19</a>, <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, , <a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/><a NAME="E16:33_1_1_23_5_3_3_1_1_3"/><b>then </b><i><font color="Green">E894</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="binarith.html#K3">-'</a> 1 &amp; <font color="Maroon">p2</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><b>set </b><font color="Maroon">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span>;<br/><a NAME="E17:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green">E895</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, </i>;<br/><a NAME="E18:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green">E896</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E19:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green">E897</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E20:33_1_1_23_5_3_3_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green">E898</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i>, <a class="ref" href="finseq_5.html#T69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan9.html#T19" target="_self">Th19</a>, , <a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>
;<br/><span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, , , <a class="ref" href="finseq_5.html#T68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>
;<br/><a NAME="E23:33_1_1_23_5_3_3_1_1_3"/><b>then </b>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">p1</font>,<span class="p1">(<span class="default"><font color="Maroon">p2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , , , <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/><a NAME="E24:33_1_1_23_5_3_3_1_1_3"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, , <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, , , </i>;<br/><a NAME="E25:33_1_1_23_5_3_3_1_1_3"/><b>then </b><i><font color="Green">E899</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E26:33_1_1_23_5_3_3_1_1_3"/><i><font color="Green">E900</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="sppol_2.html#T22">SPPOL_2:22</a></i>;<br/><a NAME="E27:33_1_1_23_5_3_3_1_1_3"/>
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D2">GOBOARD9:def 2</a></i>;<br/><b>hence </b><a NAME="E28:33_1_1_23_5_3_3_1_1_3"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T18" target="_self">Th18</a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_3_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_3_1_1_4"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_3_1_1_4"/><b>then </b><i><font color="Green">E901</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_3_1_1_4"/><b>then </b>
<font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a1</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><b>then </b> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> = 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>
<b>by </b><i>, , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , <a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>
<br/>.= 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , , , <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>
;<br/><b>hence </b><a NAME="E5:33_1_1_23_5_3_3_1_1_4"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_4"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_4"/><i><font color="Green">E902</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">a1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">a2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">p'1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">p'2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:33_1_1_23_5_3_4"/><i><font color="Green">E903</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">a2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E4:33_1_1_23_5_3_4"/><i><font color="Green">E904</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">a1</font>,<font color="Maroon">a2</font>
 <b>and </b><br/><a NAME="E5:33_1_1_23_5_3_4"/><i><font color="Green">E905</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">p'1</font>,<font color="Maroon">p'2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>and </b><br/><a NAME="E6:33_1_1_23_5_3_4"/><i><font color="Green">E906</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">p'1</font>,<font color="Maroon">p'2</font>
 <b>and </b><br/><a NAME="E7:33_1_1_23_5_3_4"/><i><font color="Green">E907</font></i>: 
( ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E8:33_1_1_23_5_3_4"/><i><font color="Green">E908</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a1</font> &amp; <font color="Maroon">a1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a2</font> &amp; <font color="Maroon">a2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><b>thus </b><a NAME="E9:33_1_1_23_5_3_4"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_5_3_4_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_1_23_5_3_4_1_1"/>
( ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> ) or ( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_4_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_4_1_1_1"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_4_1_1_1"/><b>then </b>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_4_1_1_1"/><b>then </b><i><font color="Green">E909</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> )
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i>, , , , , <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan9.html#T30" target="_self">Th30</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="jordan9.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:33_1_1_23_5_3_4_1_1_1"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T14" target="_self">Th14</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , <a class="ref" href="jordan9.html#T21" target="_self">Th21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_4_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_4_1_1_2"/><i><font color="Green">E910</font></i>: 
( <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_4_1_1_2"/><b>then </b><i><font color="Green">E911</font></i>: 
( <font color="Maroon">a1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_4_1_1_2"/><b>then </b><i><font color="Green">E912</font></i>: 
<font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q1</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E4:33_1_1_23_5_3_4_1_1_2"/><i><font color="Green">E913</font></i>: 
<font color="Maroon">a2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 <b>by </b><i>, , <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> = 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">a1</font>,<span class="p1">(<span class="default"><font color="Maroon">a2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>
<br/>.= 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , , , <a class="ref" href="jordan9.html#T31" target="_self">Th31</a>, , <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>
;<br/><b>hence </b><a NAME="E6:33_1_1_23_5_3_4_1_1_2"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_4_1_1_3"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_4_1_1_3"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_4_1_1_3"/><b>then </b><i><font color="Green">E914</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_4_1_1_3"/><b>then </b><i><font color="Green">E915</font></i>: 
<font color="Maroon">a2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> = 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>
<b>by </b><i>, , , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , , , <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>
<br/>.= 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , <a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, , <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>
;<br/><b>hence </b><a NAME="E5:33_1_1_23_5_3_4_1_1_3"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_1_23_5_3_4_1_1_4"><b>suppose </b></a><a NAME="E1:33_1_1_23_5_3_4_1_1_4"/>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:33_1_1_23_5_3_4_1_1_4"/><b>then </b><i><font color="Green">E916</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T29" target="_self">Th29</a>, , <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green">E917</font></i>: 
<font color="Maroon">p2</font> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E4:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green">E918</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/><a NAME="E5:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green">E919</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:33_1_1"><i><font color="Green">E42</font></i></a>, </i>;<br/><a NAME="E6:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green">E920</font></i>: 
<font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green">E921</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , </i>;<br/><a NAME="E8:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green">E922</font></i>: 
( <font color="Maroon">a2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">p2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">p2</font> )
 <b>by </b><i><a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><i><font color="Green">E923</font></i>: <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i>, , </i>
;<br/><a NAME="E10:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green">E924</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font>
 <b>by </b><i>, , , </i>;<br/> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> = 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">q1</font>,<font color="Maroon">q2</font>
<b>by </b><i>, , , <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, <a class="ref" href="jordan9.html#T13" target="_self">Th13</a>, <a class="ref" href="jordan9.html#T15" target="_self">Th15</a>, , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , , , <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>
<br/>.= 
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">m</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , , <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , <a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, , <a class="ref" href="jordan9.html#T31" target="_self">Th31</a>, , , , <a class="ref" href="gobrd13.html#T41">GOBRD13:41</a></i>
;<br/><a NAME="E12:33_1_1_23_5_3_4_1_1_4"/><b>then </b>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R2">turns_left</a> <font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E13:33_1_1_23_5_3_4_1_1_4"/><b>then </b>
<font color="Maroon">f</font> <a href="gobrd13.html#R2">turns_left</a> <font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="jordan9.html#T19" target="_self">Th19</a>, <a class="ref" href="jordan9.html#T32" target="_self">Th32</a>, <a class="ref" href="gobrd13.html#T45">GOBRD13:45</a></i>;<br/><a NAME="E14:33_1_1_23_5_3_4_1_1_4"/><b>then </b><i><font color="Green">E925</font></i>: 
( <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">p1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">p2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">p2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , <a class="ref" href="jordan9.html#T19" target="_self">Th19</a>, <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, <a class="ref" href="jordan9.html#T25" target="_self">Th25</a>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, , <a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/><a NAME="E15:33_1_1_23_5_3_4_1_1_4"/><b>then </b><i><font color="Green">E926</font></i>: 
<font color="Maroon">p1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><b>set </b><font color="Maroon">rc</font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span>;<br/><a NAME="E16:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green">E927</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, </i>;<br/><a NAME="E17:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green">E928</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:33_1_1_23_5_3_4_1_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green">E929</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i>, <a class="ref" href="finseq_5.html#T69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="jordan9.html#T19" target="_self">Th19</a>, , <a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>
;<br/><span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">g</font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i>, , , <a class="ref" href="finseq_5.html#T68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">g</font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#T68">FINSEQ_5:68</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>
;<br/><a NAME="E21:33_1_1_23_5_3_4_1_1_4"/><b>then </b>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">p1</font>,<font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, <a class="ref" href="jordan9.html#T22" target="_self">Th22</a>, <a class="ref" href="jordan9.html#T23" target="_self">Th23</a>, , , <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/><a NAME="E22:33_1_1_23_5_3_4_1_1_4"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T28" target="_self">Th28</a>, , <a class="ref" href="jordan9.html#T30" target="_self">Th30</a>, , </i>;<br/><a NAME="E23:33_1_1_23_5_3_4_1_1_4"/><b>then </b><i><font color="Green">E930</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="jordan9.html#T18" target="_self">Th18</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E24:33_1_1_23_5_3_4_1_1_4"/><i><font color="Green">E931</font></i>: 
 <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 <b>by </b><i>, <a class="ref" href="sppol_2.html#T22">SPPOL_2:22</a></i>;<br/><a NAME="E25:33_1_1_23_5_3_4_1_1_4"/>
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D2">GOBOARD9:def 2</a></i>;<br/><b>hence </b><a NAME="E26:33_1_1_23_5_3_4_1_1_4"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">g</font></span>)</span>
 <b>by </b><i>, , , , <a class="ref" href="jordan9.html#T18" target="_self">Th18</a>, , , </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div><b>end;</b></div><a NAME="E15:33_1_1_23"/><i><font color="Green">E932</font></i>: 
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_6"><b>proof </b></a><div class="add">

<a NAME="E1:33_1_1_23_6"/>
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , </i>;<br/>
<b>then consider </b><font color="Maroon">p</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:33_1_1_23_6"/><i><font color="Green">E933</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_23_6_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">a</font> = <font color="Maroon">p</font>;<br/><b>thus </b><a NAME="E1:33_1_1_23_6_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">u</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal3.html#T13">TOPREAL3:13</a></i>;<br/><b>consider </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:33_1_1_23_6_1"/><i><font color="Green">E935</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E5:33_1_1_23_6_1"/><i><font color="Green">E936</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">g</font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, , <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="goboard6.html#T8">GOBOARD6:8</a></i>;<br/><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><a NAME="E7:33_1_1_23_6_1"/><i><font color="Green">E937</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="goboard6.html#T4">GOBOARD6:4</a></i>;<br/><b>reconsider </b><font color="Maroon">B</font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">u</font>,<font color="Maroon">r</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i>, <a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="goboard6.html#T4">GOBOARD6:4</a>, <a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/><a NAME="E9:33_1_1_23_6_1"/><i><font color="Green">E938</font></i>: 
<font color="Maroon">B</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="sprect_3.html#T17">SPRECT_3:17</a></i>;<br/><a NAME="E10:33_1_1_23_6_1"/>
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">g</font>,1,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K4">right_cell</a> <font color="Maroon">g</font>,1
 <b>by </b><i>, , <a class="ref" href="gobrd13.html#T34">GOBRD13:34</a></i>;<br/><a NAME="E11:33_1_1_23_6_1"/><b>then </b><i><font color="Green">E939</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">g</font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K4">right_cell</a> <font color="Maroon">g</font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/><a NAME="E12:33_1_1_23_6_1"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K4">right_cell</a> <font color="Maroon">g</font>,1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="goboard9.html#T28">GOBOARD9:28</a></i>;<br/><a NAME="E13:33_1_1_23_6_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">g</font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E14:33_1_1_23_6_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">g</font>
 <b>by </b><i>, , , <a class="ref" href="gobrd13.html#T33">GOBRD13:33</a></i>;<br/><a NAME="E15:33_1_1_23_6_1"/><b>then </b><i><font color="Green">E940</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E16:33_1_1_23_6_1"/><i><font color="Green">E941</font></i>: 
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, , </i>;<br/><a NAME="E17:33_1_1_23_6_1"/>
<font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="goboard6.html#T6">GOBOARD6:6</a></i>;<br/><a NAME="E18:33_1_1_23_6_1"/><b>then </b><i><font color="Green">E942</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="jordan9.html#T11" target="_self">Th11</a>, , <a class="ref" href="tops_1.html#T39">TOPS_1:39</a></i>;<br/><a NAME="E19:33_1_1_23_6_1"/><i><font color="Green">E943</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T9" target="_self">Th9</a>, <a class="ref" href="goboard6.html#T4">GOBOARD6:4</a></i>;<br/><a NAME="E20:33_1_1_23_6_1"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="jordan9.html#T10" target="_self">Th10</a>, , , <a class="ref" href="jordan9.html#T12" target="_self">Th12</a>, <a class="ref" href="goboard9.html#T6">GOBOARD9:6</a></i>;<br/><b>hence </b><a NAME="E21:33_1_1_23_6_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="jordan9.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:33_1_1_23_6"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div><a NAME="E16:33_1_1_23"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="goboard9.html#T27">GOBOARD9:27</a></i>;<br/><a NAME="E17:33_1_1_23"/><b>then </b>
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">g</font>
 <b>by </b><i>, , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, </i>;<br/><b>hence </b><a NAME="E18:33_1_1_23"/>
contradiction
 <b>by </b><i><a class="ref" href="sprect_4.html#T7">SPRECT_4:7</a></i>;<br/></div><b>end;</b></div>
<b>then </b><i><font color="Green">E944</font></i>: <font color="Maroon">g</font> = 
<font color="Maroon">f</font> <a href="rfinseq.html#K1">/^</a> 0
<b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>
<br/>.= 
<font color="Maroon">f</font>
<b>by </b><i><a class="ref" href="finseq_5.html#T31">FINSEQ_5:31</a></i>
;<br/>
<div><i><font color="Green">E945</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_25"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:33_1_1_25"/><i><font color="Green">E946</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E3:33_1_1_25"/><i><font color="Green">E947</font></i>: 
(  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>and </b><br/><a NAME="E4:33_1_1_25"/><i><font color="Green">E948</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1"><i><font color="Green">Lemma35</font></i></a></i>;<br/><b>take </b><font color="Maroon">i</font> = <font color="Maroon">i</font>;<br/><b>thus </b><a NAME="E5:33_1_1_25"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i/>;<br/><a NAME="E6:33_1_1_25"/><i><font color="Green">E949</font></i>: 
<font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> 2
 <b>by </b><i><a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , </i>;<br/><a NAME="E7:33_1_1_25"/><i><font color="Green">E950</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> <a href="hidden.html#R1">=</a> 2
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E8:33_1_1_25"/><b>then </b>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>
;<br/><br/><a NAME="E10:33_1_1_25"/>
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span>
 <b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><b>hence </b><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> 2</span>)</span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>
;<br/><br/><b>thus </b><a NAME="E12:33_1_1_25"/>
(  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a> <b>by </b><i/>;<br/>
<a NAME="E75:33_1_1"/>
<font color="Maroon">f</font> is <a href="sprect_2.html#V1">clockwise_oriented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_26"><b>proof </b></a><div class="add">

<a NAME="E1:33_1_1_26"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 3
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E2:33_1_1_26"/><b>then </b><i><font color="Green">E951</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E3:33_1_1_26"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">k</font> &amp; <font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> holds <br/>(  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Olive">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">f</font>,<font color="Olive">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:33_1_1_26"/><b>then </b><i><font color="Green">E952</font></i>: 
 <a href="pscomp_1.html#K29">N-min</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:33_1_1_26"/>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T6">TOPREAL1:6</a></i>;<br/>
<a NAME="E6:33_1_1_26"/><b>then </b><i><font color="Green">E953</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="sppol_2.html#T15">SPPOL_2:15</a></i>;<br/>
<b>consider </b><font color="Maroon">i</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:33_1_1_26"/><i><font color="Green">E954</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>and </b><br/><a NAME="E9:33_1_1_26"/><i><font color="Green">E955</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E10:33_1_1_26"/><i><font color="Green">E956</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E11:33_1_1_26"/>
(  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i/>;<br/>
<a NAME="E12:33_1_1_26"/><i><font color="Green">E957</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T82">PSCOMP_1:82</a></i>;<br/>
<a NAME="E13:33_1_1_26"/><i><font color="Green">E958</font></i>: 
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K29">N-min</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E14:33_1_1_26"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E15:33_1_1_26"/><b>then </b><i><font color="Green">E959</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i</font>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, , , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/>
<a NAME="E16:33_1_1_26"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E17:33_1_1_26"/><b>then </b><i><font color="Green">E960</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a> 
 
<b>by </b><i>, , , <a class="ref" href="goboard5.html#T2">GOBOARD5:2</a></i>;<br/>
<a NAME="E18:33_1_1_26"/>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K15">W-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E19:33_1_1_26"/><b>then </b>
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span> <a href="euclid.html#K21">`1</a>  )
 
<b>by </b><i>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E20:33_1_1_26"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i>, , , , , <a class="ref" href="jordan9.html#T8" target="_self">Th8</a>, <a class="txt" href="jordan9.html#E10"><i><font color="Green">Lemma92</font></i></a>, <a class="ref" href="goboard7.html#T9">GOBOARD7:9</a></i>;<br/>
<a NAME="E21:33_1_1_26"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3">LSeg</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K20">NW-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K21">NE-corner</a> <span class="p3">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E22:33_1_1_26"/><b>then </b>
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K24">N-most</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">f</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E23:33_1_1_26"/>
<font color="Maroon">f</font> is <a href="sprect_2.html#V1">clockwise_oriented</a>
 <b>by </b><i>, <a class="ref" href="sprect_2.html#T34">SPRECT_2:34</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as  non <a href="funct_1.html#V3">constant</a> <a href="goboard5.html#V2">standard</a> <a href="sprect_2.html#V1">clockwise_oriented</a> <a href="goboard5.html#NM1">special_circular_sequence</a> ;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>thus </b><a NAME="E77:33_1_1"/>
<font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E78:33_1_1"/>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i</font> &amp; <font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> &amp;  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Olive">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">C</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i/>;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E79:33_1_1"/><i><font color="Green">E961</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font> )
 ;<br/>

<a NAME="E80:33_1_1"/>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 2
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E81:33_1_1"/><b>then </b><i><font color="Green">E962</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E82:33_1_1"/><b>then </b><i><font color="Green">E963</font></i>: 
<font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, </i>;<br/>
<a NAME="E83:33_1_1"/><i><font color="Green">E964</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>
<a NAME="E84:33_1_1"/><i><font color="Green">E965</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E85:33_1_1"/><i><font color="Green">E966</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i>, , , , <a class="ref" href="gobrd13.html#T43">GOBRD13:43</a></i>;<br/>
<a NAME="E86:33_1_1"/><i><font color="Green">E967</font></i>: 
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="jordan9.html#T6" target="_self">Th6</a>, , </i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_1_27"/>
(  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 ;<br/><a NAME="E2:33_1_1_27"/><b>then </b>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R1">turns_right</a> <font color="Maroon">m</font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E3:33_1_1_27"/>
<font color="Maroon">f</font> <a href="gobrd13.html#R1">turns_right</a> <font color="Maroon">m</font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="gobrd13.html#T44">GOBRD13:44</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:33_1_1_28"/>
(  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/><a NAME="E2:33_1_1_28"/><b>then </b>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R3">goes_straight</a> <font color="Maroon">m</font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , </i>;<br/><b>hence </b><a NAME="E3:33_1_1_28"/>
<font color="Maroon">f</font> <a href="gobrd13.html#R3">goes_straight</a> <font color="Maroon">m</font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="gobrd13.html#T46">GOBRD13:46</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E89:33_1_1"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E90:33_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R2">turns_left</a> <font color="Maroon">m</font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E91:33_1_1"/>
<font color="Maroon">f</font> <a href="gobrd13.html#R2">turns_left</a> <font color="Maroon">m</font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , <a class="ref" href="gobrd13.html#T45">GOBRD13:45</a></i>;<br/>


</div>
