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

<b>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E1:1_1_1"/><i><font color="Green">E2</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( ( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 implies <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></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 <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></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#NM8">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">b<sub>1</sub></font> being  <a href="struct_0.html#NM8">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">b<sub>1</sub></font> holds <br/>( (  <a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> implies ( ( <font color="Olive">b<sub>1</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> implies ( (  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> implies ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></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">b<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> implies ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></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">b<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> implies ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></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">b<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) ) ) &amp; ( (  not <font color="Olive">b<sub>1</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> or  <a href="gobrd13.html#K4">left_cell</a> <font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> ) implies <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">b<sub>1</sub></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#NM8">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="E2:1_1_1"/><i><font color="Green">E3</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a> ex <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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="E1:1_1_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> 1 or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> is   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ) or ( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> is  not   <a href="struct_0.html#NM8">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="1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_1"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>4</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
;<br/><b>thus </b><a NAME="E2:1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_1"><i><font color="Green">E4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_2"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>4</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
;<br/><b>thus </b><a NAME="E2:1_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_2"><i><font color="Green">E4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_1_1_3"/><i><font color="Green">E4</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is   <a href="struct_0.html#NM8">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">c<sub>6</sub></font> = <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM8">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="jordan13.html#E2:1_1_1_1_1_3"><i><font color="Green">E5</font></i></a></i>;<br/><b>thus </b><a NAME="E4:1_1_1_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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="E1:1_1_1_1_1_3_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1"/><i><font color="Green">E6</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:1_1_1_1_1_3_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_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="1_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="E1:1_1_1_1_1_3_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> ) or  not <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> or  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E8</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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E9</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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E10</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font> )
 <b>and </b><br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E11</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> )
 <b>and </b><br/><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E12</font></i>: 
( ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E8:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E13</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E9:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E14</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E10:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E13</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E11:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E16</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></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="E12:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E13:1_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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/><a NAME="E14:1_1_1_1_1_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E18</font></i>: 
( <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E15:1_1_1_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E19</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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>6</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E9</font></i></a></i>;<br/><b>thus </b><a NAME="E16:1_1_1_1_1_3_1_1_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_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="1_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="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1"/>
( (  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> ) or (  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> ) or  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E20</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E12</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_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">E24</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></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="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/><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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>;<br/><a NAME="E6:1_1_1_1_1_3_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,0,<font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E13</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="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></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="1_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">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/>
( <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_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">E24</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</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="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/><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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E13</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="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"/>
( ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</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="1_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">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/>
( <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_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">E24</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</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="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></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="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/><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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>;<br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font>,0 <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</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="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"/>
( ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1"/>
<font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/>
( <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_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">E24</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</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="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/><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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><i><font color="Green">E21</font></i></a>, <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</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="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"/>
( ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1"/>
<font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E20</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="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E20</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><b>thus </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1"/>
( ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E12</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></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="1_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">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_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">E24</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</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="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/><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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="binarith.html#K5">-'</a> 1</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="gobrd13.html#T35">GOBRD13:35</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E15</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="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/>
( <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_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">E24</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</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="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/><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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>8</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="gobrd13.html#T37">GOBRD13:37</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E13</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="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"/>
( ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/>
( <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_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">E24</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></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="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/><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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="gobrd13.html#T39">GOBRD13:39</a></i>;<br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,0,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E15</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="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"/>
( ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1"/>
<font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</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="1_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">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/>
( <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_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">E24</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</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="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></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="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/><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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><i><font color="Green">E21</font></i></a>, <a class="ref" href="gobrd13.html#T41">GOBRD13:41</a></i>;<br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,0 <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E13</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="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"/>
( ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1"/>
<font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E20</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="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"/><i><font color="Green">E20</font></i>: 
 <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Olive">b<sub>1</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1"/>
( ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> ) or ( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E12</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_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">E24</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</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="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/><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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="gobrd13.html#T36">GOBRD13:36</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</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="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</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="1_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">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/>
( <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_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">E24</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</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="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></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="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/><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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="gobrd13.html#T38">GOBRD13:38</a></i>;<br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font>,0 <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</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="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"/>
( ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1"/>
<font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></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="1_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">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/>
( <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_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">E24</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</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="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/><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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="gobrd13.html#T40">GOBRD13:40</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E12:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E17</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="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"/>
( ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1"/>
<font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1;<br/><b>take </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><i><font color="Green">E22</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/><i><font color="Green">E23</font></i>: 
( <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 ;<br/><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/>
( <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>6</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E4:1_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">E24</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></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="E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E5:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/><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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default">1 <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><i><font color="Green">E21</font></i></a>, <a class="ref" href="gobrd13.html#T42">GOBRD13:42</a></i>;<br/><a NAME="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,0,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="binarith.html#T51">BINARITH:51</a></i>;<br/><b>hence </b><a NAME="E7:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green">E2</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E17</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="E6:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"/>
( ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1"/>
<font color="Maroon">c<sub>11</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>11</sub></font>]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

</div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_1_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_1_1_1_2"/><i><font color="Green">E7</font></i>: 
(  not <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> or  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><b>take </b>
<font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="E2:1_1_1_1_1_3_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_1_1_1_2"><i><font color="Green">E7</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="1_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_3_1_1_2"/><i><font color="Green">E6</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>take </b>
 <a href="xboole_0.html#K1">{}</a> 
;<br/><b>thus </b><a NAME="E2:1_1_1_1_1_3_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>, <a href="xboole_0.html#K1">{}</a> ]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_3"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_1_1_3_1_1_2"><i><font color="Green">E6</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="1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_1_1_4"/><i><font color="Green">E4</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>5</sub></font> is  not   <a href="struct_0.html#NM8">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="E2:1_1_1_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>, <a href="xboole_0.html#K1">{}</a> ]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_1_1_4"><i><font color="Green">E4</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">c<sub>4</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E4:1_1_1"/><i><font color="Green">E4</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E5:1_1_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>4</sub></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="E6:1_1_1"/><i><font color="Green">E6</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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>(<a class="txt" href="jordan13.html#E2:1_1_1"><i><font color="Green">E3</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>
<a NAME="E7:1_1_1"/><i><font color="Green">E7</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="E8:1_1_1"/><b>then </b><i><font color="Green">E8</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1"><i><font color="Green">E5</font></i></a></i>;<br/>
<a NAME="E9:1_1_1"/><i><font color="Green">E9</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1_1_2"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> is   <a href="struct_0.html#NM8">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_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="E1:1_1_1_2_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>5</sub></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="1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_2_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="struct_0.html#NM8">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="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_2_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="struct_0.html#NM8">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="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_3"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>thus </b><a NAME="E2:1_1_1_2_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_2_1_3_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="struct_0.html#NM8">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="jordan13.html#E1:1_1_1_2"><i><font color="Green">E10</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_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="E1:1_1_1_2_1_3_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_3_1_1_1"/><i><font color="Green">E12</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>thus </b><a NAME="E2:1_1_1_2_1_3_1_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_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="1_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="E1:1_1_1_2_1_3_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> ) or  not <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> or  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_2_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_3_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_2_1_3_1_1_1_1_1_1"/><b>then </b>
( (  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> implies ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> implies ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> implies ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_2_1_3"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_2_1_3_1_1_1"><i><font color="Green">E12</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_2_1_3_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="struct_0.html#NM8">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="1_1_1_2_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_3_1_1_1_1_1_2"/><i><font color="Green">E13</font></i>: 
(  not <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> or  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_2_1_3_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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#NM8">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="E3:1_1_1_2_1_3_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="struct_0.html#NM8">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="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_2_1_3"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_2_1_3_1_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_2_1_3_1_1_1_1_1_2"><i><font color="Green">E13</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="1_1_1_2_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_2_1_3_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_2_1_3_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> is   <a href="struct_0.html#NM8">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="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_2_1_3"><i><font color="Green">E11</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="E10:1_1_1"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="jordan13.html#E8:1_1_1"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1"><i><font color="Green">E9</font></i></a>);<br/></i>
<a NAME="E11:1_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></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="1_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></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="E1:1_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E2:1_1_1_3"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E3:1_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> is   <a href="struct_0.html#NM8">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="jordan13.html#E4:1_1_1"><i><font color="Green">E4</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1"><i><font color="Green">E10</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:1_1_1_3"/>
<font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></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="txt" href="jordan13.html#E4:1_1_1"><i><font color="Green">E4</font></i></a>, <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="nat_1.html#NM1">Nat</a>] means  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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="E13:1_1_1"/><i><font color="Green">E11</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E14:1_1_1"/><i><font color="Green">E12</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1_1_4"/><i><font color="Green">E13</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/>

<a NAME="E2:1_1_1_4"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_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="E1:1_1_1_4_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>6</sub></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="1_1_1_4_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_4_1_1"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_4_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_4"><i><font color="Green">E14</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="1_1_1_4_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_4_1_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_4_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_4"><i><font color="Green">E14</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="1_1_1_4_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_4_1_3"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>thus </b><a NAME="E2:1_1_1_4_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_4_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_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="E1:1_1_1_4_1_3_1_1"/>
( ( <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> ) or  not <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> or  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_4_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_4_1_3_1_1_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_4_1_3_1_1_1"/><b>then </b>
( (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> implies ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> implies ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> implies ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_4"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_4_1_3"><i><font color="Green">E15</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_4_1_3_1_1_1"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_4"><i><font color="Green">E13</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="1_1_1_4_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_4_1_3_1_1_2"/>
(  not <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> or  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_4_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_4"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_4_1_3"><i><font color="Green">E15</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_4_1_3_1_1_2"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_4"><i><font color="Green">E13</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="E15:1_1_1"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="jordan13.html#E13:1_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1"><i><font color="Green">E12</font></i></a>);<br/></i>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<a NAME="E16:1_1_1"/>
2 <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="jordan1h.html#T58">JORDAN1H:58</a></i>;<br/>
<a NAME="E17:1_1_1"/><b>then </b><i><font color="Green">E14</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">c<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> holds <br/>(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> ) ) );<br/>
<a NAME="E18:1_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_5"><b>proof </b></a><div class="add">

<a NAME="E1:1_1_1_5"/>
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> ) ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> holds <br/>for <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>5</sub></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">b<sub>2</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>4</sub></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">b<sub>3</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>5</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1 ) )
 
<b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="relat_1.html#T60">RELAT_1:60</a></i>;<br/>
<b>hence </b><a NAME="E2:1_1_1_5"/>
<font color="Maroon">c<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E3:1_1_1_5"/><i><font color="Green">E16</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> )
 ;<br/>

<a NAME="E4:1_1_1_5"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></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="E5:1_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_5"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:1_1_1_5"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_1_5"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> 0</span>)</span>,<font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> 0</span>)</span>,<font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E19:1_1_1"/><i><font color="Green">E16</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1"><i><font color="Green">E1</font></i></a>, <a class="ref" href="jordan11.html#T8">JORDAN11:8</a></i>;<br/>
<a NAME="E20:1_1_1"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="jordan1h.html#T58">JORDAN1H:58</a></i>;<br/>
<a NAME="E21:1_1_1"/><b>then </b><i><font color="Green">E17</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 &amp;  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan1h.html#T58">JORDAN1H:58</a>, <a class="ref" href="sprect_3.html#T8">SPRECT_3:8</a></i>;<br/>
<a NAME="E22:1_1_1"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_6"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:1_1_1_6"/><i><font color="Green">E20</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> holds <br/>(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E3:1_1_1_6"/><i><font color="Green">E21</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E4:1_1_1_6"/><i><font color="Green">E22</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_6_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>8</sub></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="1_1_1_6_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_1"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_1_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a></i>;<br/><div><i><font color="Green">E25</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_1_1"/>
<font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E2:1_1_1_6_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_1_6_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></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="E4:1_1_1_6_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_6_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E19:1_1_1"><i><font color="Green">E16</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_1_2"/>
( <font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E2:1_1_1_6_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_1_6_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></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="E4:1_1_1_6_1_1_2"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></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">b<sub>1</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>3</sub></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">b<sub>2</sub></font> <a href="real_1.html#K5">-</a> <font color="Olive">b<sub>4</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:1_1_1_6_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E6:1_1_1_6_1_1"/><i><font color="Green">E26</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E7:1_1_1_6_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></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="E8:1_1_1_6_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></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="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E9:1_1_1_6_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E10:1_1_1_6_1_1"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_6_1_1"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_2"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a></i>;<br/><a NAME="E3:1_1_1_6_1_2"/><b>then </b><i><font color="Green">E24</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/><a NAME="E4:1_1_1_6_1_2"/><i><font color="Green">E25</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1"><i><font color="Green">E1</font></i></a>, <a class="ref" href="jordan11.html#T8">JORDAN11:8</a>, <a class="ref" href="jordan11.html#T9">JORDAN11:9</a></i>;<br/><div><i><font color="Green">E26</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_2_1"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><a NAME="E2:1_1_1_6_1_2_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 )
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_1_6_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>9</sub></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="E4:1_1_1_6_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_2_1"><i><font color="Green">E27</font></i></a>, <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="1_1_1_6_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_2_2"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E2:1_1_1_6_1_2_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 2 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> 2 )
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_1_6_1_2_2"/><b>then </b><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>9</sub></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">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E4:1_1_1_6_1_2_2"/><i><font color="Green">E29</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> )
 ;<br/><a NAME="E5:1_1_1_6_1_2_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_2_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_2_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_2_2"><i><font color="Green">E29</font></i></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="E6:1_1_1_6_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E21:1_1_1"><i><font color="Green">E17</font></i></a>, <a class="ref" href="jordan3.html#T6">JORDAN3:6</a></i>;<br/><a NAME="E7:1_1_1_6_1_2_2"/><b>then </b>
( <font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> 0 )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_6_1_2_2"><i><font color="Green">E30</font></i></a></i>;<br/><a NAME="E8:1_1_1_6_1_2_2"/><b>then </b>
(  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></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="E9:1_1_1_6_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">c<sub>10</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>13</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:1_1_1_6_1_2"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_6_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/><b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E8:1_1_1_6_1_2"/><i><font color="Green">E28</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><a NAME="E9:1_1_1_6_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">c<sub>9</sub></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="E10:1_1_1_6_1_2"/><b>then </b>
<font color="Maroon">c<sub>9</sub></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="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_2"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E11:1_1_1_6_1_2"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E12:1_1_1_6_1_2"/><i><font color="Green">E30</font></i>: 
( <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E17:1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="jordan5b.html#T1">JORDAN5B:1</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E13:1_1_1_6_1_2"/><i><font color="Green">E31</font></i>: 
 <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <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="E14:1_1_1_6_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_6_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="gobrd13.html#D2">GOBRD13:def 2</a></i>;<br/><a NAME="E15:1_1_1_6_1_2"/><i><font color="Green">E33</font></i>: 
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_2"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_2"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_6_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="gobrd13.html#D3">GOBRD13:def 3</a></i>;<br/><b>thus </b><a NAME="E16:1_1_1_6_1_2"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1"><i><font color="Green">E1</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan11.html#T11">JORDAN11:11</a></i>;<br/><b>thus </b><a NAME="E17:1_1_1_6_1_2"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1"><i><font color="Green">E1</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_6_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan11.html#T10">JORDAN11:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_1_3"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E3:1_1_1_6_1_3"/><b>then </b><i><font color="Green">E24</font></i>: 
not <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><a NAME="E4:1_1_1_6_1_3"/><i><font color="Green">E25</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E5:1_1_1_6_1_3"/><i><font color="Green">E26</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:1_1_1_6_1_3"/><i><font color="Green">E27</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> )
 <b>and </b><br/><a NAME="E8:1_1_1_6_1_3"/><i><font color="Green">E28</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E9:1_1_1_6_1_3"/>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E10:1_1_1_6_1_3"/><i><font color="Green">E29</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E11:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/><a NAME="E12:1_1_1_6_1_3"/><b>then </b><i><font color="Green">E30</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:1_1_1_6_1_3"/><i><font color="Green">E31</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E14:1_1_1_6_1_3"/><i><font color="Green">E32</font></i>: 
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a></i>;<br/><a NAME="E15:1_1_1_6_1_3"/><b>then </b><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan1h.html#T61">JORDAN1H:61</a></i>;<br/><a NAME="E16:1_1_1_6_1_3"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan1h.html#T62">JORDAN1H:62</a></i>;<br/><a NAME="E17:1_1_1_6_1_3"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan1h.html#T63">JORDAN1H:63</a></i>;<br/><a NAME="E18:1_1_1_6_1_3"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></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">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan1h.html#T64">JORDAN1H:64</a></i>;<br/><a NAME="E19:1_1_1_6_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 2
 ;<br/><a NAME="E20:1_1_1_6_1_3"/><b>then </b><i><font color="Green">E37</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan1h.html#T65">JORDAN1H:65</a></i>;<br/><a NAME="E21:1_1_1_6_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 2
 ;<br/><a NAME="E22:1_1_1_6_1_3"/><b>then </b><i><font color="Green">E38</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan1h.html#T66">JORDAN1H:66</a></i>;<br/><a NAME="E23:1_1_1_6_1_3"/><i><font color="Green">E39</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan1h.html#T67">JORDAN1H:67</a></i>;<br/><a NAME="E24:1_1_1_6_1_3"/><i><font color="Green">E40</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan1h.html#T68">JORDAN1H:68</a></i>;<br/><a NAME="E25:1_1_1_6_1_3"/><i><font color="Green">E41</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan1h.html#T69">JORDAN1H:69</a></i>;<br/><a NAME="E26:1_1_1_6_1_3"/><i><font color="Green">E42</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan1h.html#T70">JORDAN1H:70</a></i>;<br/><a NAME="E27:1_1_1_6_1_3"/><i><font color="Green">E43</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan1h.html#T71">JORDAN1H:71</a></i>;<br/><a NAME="E28:1_1_1_6_1_3"/><i><font color="Green">E44</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></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">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="jordan1h.html#T72">JORDAN1H:72</a></i>;<br/><b>thus </b><a NAME="E29:1_1_1_6_1_3"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_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="E1:1_1_1_6_1_3_1_1"/>
( (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> ) or (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> ) or  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_1_1_1"/>
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_1_1_1"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_1_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><a NAME="E5:1_1_1_6_1_3_1_1_1"/><i><font color="Green">E48</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_6_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E6:1_1_1_6_1_3_1_1_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/><b>thus </b><a NAME="E7:1_1_1_6_1_3_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_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="E1:1_1_1_6_1_3_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_1_1_1"><i><font color="Green">E48</font></i></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="1_1_1_6_1_3_1_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_1_1_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_1_1_1_1"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_1_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_6_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_6_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:1_1_1_6_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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></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">c<sub>16</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_1_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></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="E4:1_1_1_6_1_3_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_6_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_1_1_1_1"><i><font color="Green">E51</font></i></a>, <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="1_1_1_6_1_3_1_1_1_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_1_1_1_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_1_1_1_2"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></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="1_1_1_6_1_3_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_1_1_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_1_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_6_1_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_1_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></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="E4:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_6_1_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_1_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_1_1_1_2"><i><font color="Green">E51</font></i></a>, <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="1_1_1_6_1_3_1_1_1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_1_1_1_3"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_1_1_1_3"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_1_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_1_1_1_3_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_1_1_1_3_1"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_6_1_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_1_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_1_1_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_6_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_1_1_1_3_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></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="E4:1_1_1_6_1_3_1_1_1_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_6_1_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_1_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_1_1_1_3"><i><font color="Green">E51</font></i></a>, <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="1_1_1_6_1_3_1_1_1_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_1_1_1_4"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_1_1_1_4"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_1_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_1_1_1_4_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_1_1_1_4_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_6_1_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_1_1_1_4"><i><font color="Green">E50</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></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="E4:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_1_1_1_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_6_1_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_1_1_1_4"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_1_1_1_4"><i><font color="Green">E51</font></i></a>, <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="1_1_1_6_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_1_1_2"/><i><font color="Green">E46</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_1_1_2"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><a NAME="E5:1_1_1_6_1_3_1_1_2"/><i><font color="Green">E49</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_6_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E6:1_1_1_6_1_3_1_1_2"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/><b>thus </b><a NAME="E7:1_1_1_6_1_3_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_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="E1:1_1_1_6_1_3_1_1_2_1_1"/>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_1_1_2"><i><font color="Green">E49</font></i></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="1_1_1_6_1_3_1_1_2_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_2_1_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_2_1_1_1"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></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="1_1_1_6_1_3_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_2_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_2_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_6_1_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></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="E4:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_6_1_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_2_1_1_1"><i><font color="Green">E52</font></i></a>, <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="1_1_1_6_1_3_1_1_2_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_2_1_1_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_2_1_1_2"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_2_1_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_2_1_1_2_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_6_1_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></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="E4:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_6_1_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_2_1_1_2"><i><font color="Green">E52</font></i></a>, <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="1_1_1_6_1_3_1_1_2_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_2_1_1_3"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_2_1_1_3"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_2_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_2_1_1_3_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_2_1_1_3_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_6_1_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2_1_1_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_2_1_1_3_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_6_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:1_1_1_6_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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></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">c<sub>16</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_2_1_1_3_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></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="E4:1_1_1_6_1_3_1_1_2_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_6_1_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2_1_1_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <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="1_1_1_6_1_3_1_1_2_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_2_1_1_4"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_2_1_1_4"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_2_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_2_1_1_4_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_2_1_1_4_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_6_1_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2_1_1_4"><i><font color="Green">E51</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_2_1_1_4_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_6_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_2_1_1_4_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></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="E4:1_1_1_6_1_3_1_1_2_1_1_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_6_1_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_2_1_1_4"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_2_1_1_4"><i><font color="Green">E52</font></i></a>, <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="1_1_1_6_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_1_1_3"/><i><font color="Green">E46</font></i>: 
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_1_1_3"/><i><font color="Green">E47</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_1_1_3"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>;<br/><a NAME="E5:1_1_1_6_1_3_1_1_3"/><i><font color="Green">E49</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_6_1_3"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E6:1_1_1_6_1_3_1_1_3"/><i><font color="Green">E50</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/><b>thus </b><a NAME="E7:1_1_1_6_1_3_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_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="E1:1_1_1_6_1_3_1_1_3_1_1"/>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_1_1_3"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_1_1_3"><i><font color="Green">E49</font></i></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="1_1_1_6_1_3_1_1_3_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_3_1_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_3_1_1_1"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_3_1_1_1_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_3_1_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_6_1_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></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="E4:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_6_1_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_3"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_3_1_1_1"><i><font color="Green">E52</font></i></a>, <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="1_1_1_6_1_3_1_1_3_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_3_1_1_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_3_1_1_2"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_3_1_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_3_1_1_2_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_3_1_1_2_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_6_1_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_3_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_6_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>17</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_3_1_1_2_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></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="E4:1_1_1_6_1_3_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_6_1_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_3"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_3_1_1_2"><i><font color="Green">E52</font></i></a>, <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="1_1_1_6_1_3_1_1_3_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_3_1_1_3"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_3_1_1_3"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></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="1_1_1_6_1_3_1_1_3_1_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_3_1_1_3_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_3_1_1_3_1"/><b>then </b>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1_6_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3_1_1_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></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="E4:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_6_1_3_1_1_3_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1_6_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_3"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3_1_1_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_3_1_1_3"><i><font color="Green">E52</font></i></a>, <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="1_1_1_6_1_3_1_1_3_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_1_1_3_1_1_4"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_1_1_3_1_1_4"/><i><font color="Green">E52</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_1_1_3_1_1_4_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:1_1_1_6_1_3_1_1_3_1_1_4_1"/>
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font> )
 ;<br/><a NAME="E2:1_1_1_6_1_3_1_1_3_1_1_4_1"/><b>then </b><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3_1_1_4"><i><font color="Green">E51</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_1_1_3_1_1_4_1"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="real_1.html#K5">-</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_6_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/><a NAME="E4:1_1_1_6_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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></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">c<sub>16</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_3_1_1_4_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_6_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">c<sub>18</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>16</sub></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">c<sub>19</sub></font> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>17</sub></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="E4:1_1_1_6_1_3_1_1_3_1_1_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_1_1_3"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_1_1_3_1_1_4"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_1_1_3_1_1_4"><i><font color="Green">E52</font></i></a>, <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">c<sub>13</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E30:1_1_1_6_1_3"/><i><font color="Green">E46</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_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="E1:1_1_1_6_1_3_2_1"/>
( <font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> or <font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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="1_1_1_6_1_3_2_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_1"/>
<font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_1_3_2_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green">E22</font></i></a></i>;<br/><a NAME="E3:1_1_1_6_1_3_2_1_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_6_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E4:1_1_1_6_1_3_2_1_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_6_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><b>thus </b><a NAME="E5:1_1_1_6_1_3_2_1_1"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_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="E1:1_1_1_6_1_3_2_1_1_1_1"/>
( (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> ) or (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> ) or  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_1"/><i><font color="Green">E50</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_1"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_1"/><i><font color="Green">E53</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_6_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green">E52</font></i></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="1_1_1_6_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="1_1_1_6_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="E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green">E53</font></i></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="1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T35">GOBRD13:35</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_6_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green">E54</font></i></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="E6:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_6_1_3"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T37">GOBRD13:37</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_6_1_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green">E54</font></i></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="E6:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_6_1_3"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T39">GOBRD13:39</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_6_1_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/><a NAME="E5:1_1_1_6_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_6_1_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T41">GOBRD13:41</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_6_1_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/><a NAME="E5:1_1_1_6_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_6_1_3"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_1_1_1_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:1_1_1_6_1_3_2_1_1_1_1_1"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_2"/><i><font color="Green">E50</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_2"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_2"/><i><font color="Green">E53</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_6_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E52</font></i></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="1_1_1_6_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="1_1_1_6_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="E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1"/>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E53</font></i></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="1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T36">GOBRD13:36</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_6_1_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/><a NAME="E5:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T35">GOBRD13:35</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_6_1_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T38">GOBRD13:38</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_6_1_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/><a NAME="E5:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T37">GOBRD13:37</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_6_1_3"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T40">GOBRD13:40</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_6_1_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/><a NAME="E5:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T39">GOBRD13:39</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_6_1_3"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T42">GOBRD13:42</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_6_1_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/><a NAME="E5:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T41">GOBRD13:41</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_6_1_3"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_2_1_1_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:1_1_1_6_1_3_2_1_1_1_1_2"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_3"/><i><font color="Green">E50</font></i>: 
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_3"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_2_1_1_1_1_3"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_3"/><i><font color="Green">E53</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_6_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E52</font></i></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="1_1_1_6_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="1_1_1_6_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="E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1"/>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_6_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E53</font></i></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="1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green">E54</font></i></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="E4:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_6_1_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T36">GOBRD13:36</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_6_1_3"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green">E54</font></i></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="E4:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_6_1_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T38">GOBRD13:38</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_6_1_3"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1_6_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T40">GOBRD13:40</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1_6_1_3"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_3"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"/><i><font color="Green">E55</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E3:1_1_1_6_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/><a NAME="E5:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6_1_3"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_6_1_3"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T42">GOBRD13:42</a></i>;<br/><b>hence </b><a NAME="E6:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_6_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1_6_1_3"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_1_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_6_1_3_2_1_1_1_1_3_1_1_4"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:1_1_1_6_1_3_2_1_1_1_1_3"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></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="1_1_1_6_1_3_2_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_2"/>
<font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_1_3_2_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:1_1_1_6_1_3_2_1_2"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_6"><i><font color="Green">E22</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:1_1_1_6_1_3_2_1_2"/><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> )
 <b>and </b><br/><a NAME="E6:1_1_1_6_1_3_2_1_2"/><i><font color="Green">E49</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>and </b><br/><a NAME="E7:1_1_1_6_1_3_2_1_2"/><i><font color="Green">E50</font></i>: 
( ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E8:1_1_1_6_1_3_2_1_2"/><i><font color="Green">E51</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></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="E9:1_1_1_6_1_3_2_1_2"/>
<font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:1_1_1_6_1_3_2_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_6_1_3_2_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:1_1_1_6_1_3_2_1_2"/><i><font color="Green">E53</font></i>: 
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_6"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_2"><i><font color="Green">E47</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_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="E1:1_1_1_6_1_3_2_1_2_1_1"/>
( (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> ) or (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> ) or  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_2_1_1_1"/>
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_2_1_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_2_1_2_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font>;<br/><b>thus </b><a NAME="E5:1_1_1_6_1_3_2_1_2_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_2_1_1_1"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_2_1_1_2"/>
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_2_1_2_1_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font>;<br/><b>thus </b><a NAME="E5:1_1_1_6_1_3_2_1_2_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_2_1_1_2"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_2_1_1_3"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_1_6_1_3_2_1_2_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:1_1_1_6_1_3_2_1_2_1_1_3"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3"><i><font color="Green">E32</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>19</sub></font>;<br/><b>thus </b><a NAME="E5:1_1_1_6_1_3_2_1_2_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_6_1_3_2_1_2_1_1_3"><i><font color="Green">E54</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:1_1_1_6_1_3_2_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 ;<br/><a NAME="E15:1_1_1_6_1_3_2_1_2"/><i><font color="Green">E55</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>13</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_6_1_3_2_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_6_1_3_2_1_2"><i><font color="Green">E54</font></i></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="1_1_1_6_1_3_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_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="E1:1_1_1_6_1_3_2_1_2_2_1"/>
( ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_6_1_3_2_1_2"><i><font color="Green">E50</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_6_1_3_2_1_2_2_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_2_2_1_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>15</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green">E49</font></i></a>, <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="E3:1_1_1_6_1_3_2_1_2_2_1_1"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_3_2_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_6_1_3_2_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_2_2_1_1"><i><font color="Green">E56</font></i></a>, <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="1_1_1_6_1_3_2_1_2_2_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_2_2_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>15</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green">E49</font></i></a>, <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="E3:1_1_1_6_1_3_2_1_2_2_1_2"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_3_2_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_6_1_3_2_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_2_2_1_2"><i><font color="Green">E56</font></i></a>, <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="1_1_1_6_1_3_2_1_2_2_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_2_2_1_3"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green">E49</font></i></a>, <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="E3:1_1_1_6_1_3_2_1_2_2_1_3"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_3_2_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_6_1_3_2_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_2_2_1_3"><i><font color="Green">E56</font></i></a>, <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="1_1_1_6_1_3_2_1_2_2_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_6_1_3_2_1_2_2_1_4"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_6_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>17</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_6"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_6_1_3_2_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green">E49</font></i></a>, <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="E3:1_1_1_6_1_3_2_1_2_2_1_4"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1_6_1_3"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_6_1_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_6_1_3_2_1_2"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_6_1_3_2_1_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_6_1_3_2_1_2"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_6_1_3_2_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_6_1_3_2_1_2_2_1_4"><i><font color="Green">E56</font></i></a>, <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="E17:1_1_1_6_1_3_2_1_2"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E32:1_1_1_6_1_3"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E23:1_1_1"/><i><font color="Green">E19</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="jordan13.html#E18:1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1"><i><font color="Green">E18</font></i></a>);<br/></i>
<a NAME="E24:1_1_1"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0,<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> 0,<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span>]
 
<b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E25:1_1_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[1,<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> 1,<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.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="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a></i>;<br/>
<a NAME="E26:1_1_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Olive">b<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Olive">b<sub>5</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; ( <font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Olive">b<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Olive">b<sub>5</sub></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">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></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">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">b<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">b<sub>5</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_7"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:1_1_1_7"/><i><font color="Green">E24</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E3:1_1_1_7"/><i><font color="Green">E25</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 ;<br/>

<a NAME="E4:1_1_1_7"/><i><font color="Green">E26</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/>
<a NAME="E5:1_1_1_7"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E6:1_1_1_7"/><i><font color="Green">E28</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E7:1_1_1_7"/><b>then </b><i><font color="Green">E29</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_7"><i><font color="Green">E23</font></i></a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/>
<a NAME="E8:1_1_1_7"/><i><font color="Green">E30</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_7"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_7"><i><font color="Green">E28</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E9:1_1_1_7"/><b>then </b><i><font color="Green">E31</font></i>: 
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_7"><i><font color="Green">E29</font></i></a></i>;<br/>
<a NAME="E10:1_1_1_7"/><i><font color="Green">E32</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_7"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_7"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_7"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_7"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:1_1_1_7"/><i><font color="Green">E33</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_7"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E12:1_1_1_7"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>12</sub></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="E1:1_1_1_7_1"/>
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_1_7_1"/><i><font color="Green">E35</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:1_1_1_7_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_7"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_7"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_7"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_7"><i><font color="Green">E31</font></i></a></i>;<br/><a NAME="E5:1_1_1_7_1"/><i><font color="Green">E37</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_7"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_7"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_7_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E6:1_1_1_7_1"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_7_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/><b>thus </b><a NAME="E7:1_1_1_7_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_7_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_7_1"><i><font color="Green">E36</font></i></a></i>;<br/><b>thus </b><a NAME="E8:1_1_1_7_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_7"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_7"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_7"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_7"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_7_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_7_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_7_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/><b>thus </b><a NAME="E9:1_1_1_7_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></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="txt" href="jordan13.html#E2:1_1_1_7"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_7"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_7"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_7"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_7_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_7_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_7_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/><b>thus </b><a NAME="E10:1_1_1_7_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_7"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_7"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_7"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_7"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_7_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_7_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_7_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/><b>thus </b><a NAME="E11:1_1_1_7_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></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">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_7"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_7"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_7"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_7"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_7_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_7_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_7_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_7_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E27:1_1_1"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Olive">b<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Olive">b<sub>5</sub></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">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">b<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">b<sub>5</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Olive">b<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Olive">b<sub>5</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></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">b<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_8"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:1_1_1_8"/><i><font color="Green">E25</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E3:1_1_1_8"/><i><font color="Green">E26</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 ;<br/>

<a NAME="E4:1_1_1_8"/><i><font color="Green">E27</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/>
<a NAME="E5:1_1_1_8"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E6:1_1_1_8"/><i><font color="Green">E29</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E7:1_1_1_8"/><b>then </b><i><font color="Green">E30</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_8"><i><font color="Green">E24</font></i></a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/>
<a NAME="E8:1_1_1_8"/><i><font color="Green">E31</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_8"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_8"><i><font color="Green">E29</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E9:1_1_1_8"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_8"><i><font color="Green">E30</font></i></a></i>;<br/>
<a NAME="E10:1_1_1_8"/><i><font color="Green">E33</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_8"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_8"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_8"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_8"><i><font color="Green">E30</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:1_1_1_8"/><i><font color="Green">E34</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_8"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E12:1_1_1_8"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>12</sub></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="E1:1_1_1_8_1"/>
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_1_8_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E4:1_1_1_8_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_8"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_8"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_8"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_8"><i><font color="Green">E32</font></i></a></i>;<br/><a NAME="E5:1_1_1_8_1"/><i><font color="Green">E38</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_8"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_8"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_8_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><a NAME="E6:1_1_1_8_1"/><i><font color="Green">E39</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_8_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/><b>thus </b><a NAME="E7:1_1_1_8_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_8_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_8_1"><i><font color="Green">E37</font></i></a></i>;<br/><b>thus </b><a NAME="E8:1_1_1_8_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></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="txt" href="jordan13.html#E2:1_1_1_8"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_8"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_8"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_8"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_8_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_8_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_8_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E9:1_1_1_8_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_8"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_8"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_8"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_8"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_8_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_8_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_8_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E10:1_1_1_8_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_8"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_8"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_8"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_8"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_8_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_8_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_8_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a></i>;<br/><b>thus </b><a NAME="E11:1_1_1_8_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></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">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_8"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_8"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_8"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_8"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_8_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_8_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_8_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_8_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a></i>;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E28:1_1_1"/><i><font color="Green">E24</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font>, <font color="Olive">b<sub>5</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>5</sub></font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">b<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Olive">b<sub>5</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; ( <font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) &amp; ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<span class="p2">(<span class="default"><font color="Olive">b<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font>,<span class="p3">(<span class="default"><font color="Olive">b<sub>5</sub></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">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></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">b<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Olive">b<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Olive">b<sub>5</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_9"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>









<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_9"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 <b>and </b><br/><a NAME="E2:1_1_1_9"/><i><font color="Green">E26</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E3:1_1_1_9"/><i><font color="Green">E27</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 ;<br/>

<a NAME="E4:1_1_1_9"/><i><font color="Green">E28</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/>
<a NAME="E5:1_1_1_9"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E6:1_1_1_9"/><i><font color="Green">E30</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E7:1_1_1_9"/><b>then </b><i><font color="Green">E31</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_9"><i><font color="Green">E25</font></i></a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/>
<a NAME="E8:1_1_1_9"/><i><font color="Green">E32</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_9"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_9"><i><font color="Green">E30</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E9:1_1_1_9"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_9"><i><font color="Green">E31</font></i></a></i>;<br/>
<a NAME="E10:1_1_1_9"/><i><font color="Green">E34</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_9"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_9"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_9"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_9"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:1_1_1_9"/><i><font color="Green">E35</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_9"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E12:1_1_1_9"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>assume </b><a NAME="E13:1_1_1_9"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E15:1_1_1_9"/><i><font color="Green">E37</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E16:1_1_1_9"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_9"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_9"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_9"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_9"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E17:1_1_1_9"/><i><font color="Green">E39</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_9"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_9"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_9"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_9"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/>
<a NAME="E18:1_1_1_9"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E16:1_1_1_9"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_4.html#T82">FINSEQ_4:82</a></i>;<br/>
<b>thus </b><a NAME="E19:1_1_1_9"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1_9"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_9"><i><font color="Green">E38</font></i></a></i>;<br/>

<b>thus </b><a NAME="E20:1_1_1_9"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_9"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_9"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_9"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_9"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_9"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_9"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_9"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_9"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_9"><i><font color="Green">E40</font></i></a>, <a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/>

<b>thus </b><a NAME="E21:1_1_1_9"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_9"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_9"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_9"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_9"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_9"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_9"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_9"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_9"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_9"><i><font color="Green">E40</font></i></a>, <a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/>

<b>thus </b><a NAME="E22:1_1_1_9"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> implies ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></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="txt" href="jordan13.html#E2:1_1_1_9"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_9"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_9"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_9"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_9"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_9"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_9"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_9"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_9"><i><font color="Green">E40</font></i></a>, <a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/>

<b>thus </b><a NAME="E23:1_1_1_9"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></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">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_9"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_9"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_9"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_9"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_9"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_9"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_9"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_9"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_9"><i><font color="Green">E40</font></i></a>, <a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:1_1_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:1_1_1_10"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E2:1_1_1_10"/><i><font color="Green">E27</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_10_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1 or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>8</sub></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="1_1_1_10_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_1_1"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>8</sub></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>take </b>
 <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
;<br/><b>take </b>
 <a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
;<br/><b>thus </b><a NAME="E3:1_1_1_10_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1"><i><font color="Green">E1</font></i></a>, <a class="ref" href="jordan11.html#T8">JORDAN11:8</a></i>;<br/><b>thus </b><a NAME="E4:1_1_1_10_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_10_1_1"><i><font color="Green">E28</font></i></a>, <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="1_1_1_10_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_2"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>take </b>
<span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
;<br/><b>take </b>
 <a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
;<br/><b>thus </b><a NAME="E2:1_1_1_10_1_2"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1"><i><font color="Green">E1</font></i></a>, <a class="ref" href="jordan11.html#T9">JORDAN11:9</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_10_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E24:1_1_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_2"><i><font color="Green">E28</font></i></a>, <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="1_1_1_10_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_1_3"/><b>then </b><i><font color="Green">E29</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_10"><i><font color="Green">E27</font></i></a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E3:1_1_1_10_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_10"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green">E28</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:1_1_1_10_1_3"/><i><font color="Green">E30</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font> )
 <b>and </b><br/><a NAME="E6:1_1_1_10_1_3"/><i><font color="Green">E31</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font> )
 <b>and </b><br/><a NAME="E7:1_1_1_10_1_3"/><i><font color="Green">E32</font></i>: 
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_10"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_10_1_3"><i><font color="Green">E29</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_10_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_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="E1:1_1_1_10_1_3_1_1"/>
( (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> ) or (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> ) or  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_1"/><i><font color="Green">E33</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_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="E1:1_1_1_10_1_3_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_10_1_3"><i><font color="Green">E32</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_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">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_1"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_1_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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="1_1_1_10_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_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">c<sub>11</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></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="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_1"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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="1_1_1_10_1_3_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_1_1_1_3"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_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">c<sub>11</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_1"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_1_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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="1_1_1_10_1_3_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_1_1_1_4"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_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">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_1"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_1_1_1_4"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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="E3:1_1_1_10_1_3_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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="1_1_1_10_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_2"/><i><font color="Green">E33</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_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="E1:1_1_1_10_1_3_1_1_2_1_1"/>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_10_1_3"><i><font color="Green">E32</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_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">c<sub>11</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></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="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_2"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_2_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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="1_1_1_10_1_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_2_1_1_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_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">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_2"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_2_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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="1_1_1_10_1_3_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_2_1_1_3"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_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">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_2"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_2_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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="1_1_1_10_1_3_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_2_1_1_4"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_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">c<sub>11</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_2"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_2_1_1_4"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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="E3:1_1_1_10_1_3_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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="1_1_1_10_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_3"/><i><font color="Green">E33</font></i>: 
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_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="E1:1_1_1_10_1_3_1_1_3_1_1"/>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_10_1_3"><i><font color="Green">E32</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_10_1_3_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_3_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_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">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_3"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_3_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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="1_1_1_10_1_3_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_3_1_1_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_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">c<sub>11</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_3"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_3_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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="1_1_1_10_1_3_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_3_1_1_3"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_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">c<sub>11</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></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="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_3"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_3_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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="1_1_1_10_1_3_1_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_10_1_3_1_1_3_1_1_4"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_10_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">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_10_1_3"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_10_1_3"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_10_1_3_1_1_3"><i><font color="Green">E33</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_10_1_3_1_1_3_1_1_4"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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="E3:1_1_1_10_1_3_1_1_3"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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="E9:1_1_1_10_1_3"/>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></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">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>;<br/>
<a NAME="E30:1_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">S<sub>5</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1_1_11"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:1_1_1_11"/>
0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_11"><i><font color="Green">E27</font></i></a></i>;<br/>
<a NAME="E3:1_1_1_11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="relat_1.html#K7">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K2">Seg</a> 0</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D15">FINSEQ_1:def 15</a></i>;<br/>
<a NAME="E4:1_1_1_11"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="relat_1.html#T110">RELAT_1:110</a></i>;<br/>
<b>hence </b><a NAME="E5:1_1_1_11"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1"><i><font color="Green">E5</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_11"><i><font color="Green">E27</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:1_1_1"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_12"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1_1_12"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E2:1_1_1_12"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_12_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_12"><i><font color="Green">E29</font></i></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="1_1_1_12_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_12_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_1_12_1_1"/><b>then </b><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:1_1_1_12_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:1_1_1_12_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:1_1_1_12_1_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E7:1_1_1_12_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_12_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_12_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_12_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="finseq_5.html#T25">FINSEQ_5:25</a></i>;<br/><b>hence </b><a NAME="E8:1_1_1_12_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_12"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_12_1_1"><i><font color="Green">E30</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_12_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_12_1_2"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_1_12_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_12_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_12_1_2"><i><font color="Green">E30</font></i></a>, <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="E32:1_1_1"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="jordan13.html#E30:1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan13.html#E31:1_1_1"><i><font color="Green">E27</font></i></a>);<br/></i>
<a NAME="E33:1_1_1"/><i><font color="Green">E29</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 holds <br/>( (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R2">turns_left</a> <font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) &amp; (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R3">goes_straight</a> <font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) &amp; (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R1">turns_right</a> <font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_13"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1_1_13"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/>

<a NAME="E2:1_1_1_13"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E3:1_1_1_13"/><i><font color="Green">E32</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E4:1_1_1_13"/><b>then </b><i><font color="Green">E33</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_13"><i><font color="Green">E30</font></i></a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/>
<a NAME="E5:1_1_1_13"/>
<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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_13"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_13"><i><font color="Green">E32</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E6:1_1_1_13"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font> &amp; ( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> ) or ( <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> ) or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> ) or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1 ) ) )
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_13"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_13"><i><font color="Green">E33</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_1_13"/>
( (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R2">turns_left</a> <font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) &amp; (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R3">goes_straight</a> <font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) &amp; (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R1">turns_right</a> <font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_13"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_13"><i><font color="Green">E32</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>6</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>5</sub></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="E34:1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E35:1_1_1"/><b>then </b><i><font color="Green">E30</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="E36:1_1_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>6</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_14"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1_1_14"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> is <a href="topreal1.html#V2">unfolded</a>
 ;<br/>

<a NAME="E2:1_1_1_14"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_14_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 or <font color="Maroon">c<sub>8</sub></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="1_1_1_14_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></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="E3:1_1_1_14_1_1"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_14_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="1_1_1_14_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1;<br/><a NAME="E2:1_1_1_14_1_2"/><i><font color="Green">E35</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E3:1_1_1_14_1_2"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E4:1_1_1_14_1_2"/><i><font color="Green">E37</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font>, <font color="Maroon">c<sub>13</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:1_1_1_14_1_2"/><i><font color="Green">E38</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E7:1_1_1_14_1_2"/><i><font color="Green">E39</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font> )
 <b>and </b><br/><a NAME="E8:1_1_1_14_1_2"/><i><font color="Green">E40</font></i>: 
( ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_14"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E9:1_1_1_14_1_2"/><i><font color="Green">E41</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E10:1_1_1_14_1_2"/><i><font color="Green">E42</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E11:1_1_1_14_1_2"/><b>then </b><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E12:1_1_1_14_1_2"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E13:1_1_1_14_1_2"/><i><font color="Green">E45</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_14_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_14_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_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="E1:1_1_1_14_1_2_1_1"/>
( (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> ) or (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> ) or  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_1"/><i><font color="Green">E46</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_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="E1:1_1_1_14_1_2_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_14_1_2"><i><font color="Green">E40</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_1_1_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_1"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_1_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_14_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_1_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard7.html#T18">GOBOARD7:18</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_1_1_1_1"><i><font color="Green">E48</font></i></a>, <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="1_1_1_14_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_1_1_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></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="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_1"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_1_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>13</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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="E4:1_1_1_14_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_1_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard7.html#T20">GOBOARD7:20</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_1_1_1_2"><i><font color="Green">E48</font></i></a>, <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="1_1_1_14_1_2_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_1_1_1_3"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_1"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_1_1_1_3"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_14_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_1_1_1_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard7.html#T17">GOBOARD7:17</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_1_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_1_1_1_3"><i><font color="Green">E48</font></i></a>, <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="1_1_1_14_1_2_1_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_1_1_1_4"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_1_1_1_4"/><b>then </b><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_1"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_1_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_1_1_1_4"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard7.html#T19">GOBOARD7:19</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_1_1_1_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_1_1_1_4"><i><font color="Green">E48</font></i></a>, <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="E3:1_1_1_14_1_2_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="1_1_1_14_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_2"/><i><font color="Green">E46</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_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="E1:1_1_1_14_1_2_1_1_2_1_1"/>
( ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_14_1_2"><i><font color="Green">E40</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_2_1_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></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="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_2_1_1_1"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>13</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard7.html#T15">GOBOARD7:15</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_2_1_1_1"><i><font color="Green">E48</font></i></a>, <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="1_1_1_14_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_2_1_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_2_1_1_2"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></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="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard7.html#T16">GOBOARD7:16</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_2_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <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="1_1_1_14_1_2_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_2_1_1_3"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_2_1_1_3"/><b>then </b><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_2_1_1_3"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</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">c<sub>12</sub></font> <a href="binarith.html#K5">-'</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="E4:1_1_1_14_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_14_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2_1_1_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard7.html#T16">GOBOARD7:16</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_2_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_2_1_1_3"><i><font color="Green">E48</font></i></a>, <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="1_1_1_14_1_2_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_2_1_1_4"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_2_1_1_4"/><b>then </b><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E27:1_1_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_2_1_1_4"/><b>then </b>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</a> 1 &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</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">c<sub>13</sub></font> <a href="binarith.html#K5">-'</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="E4:1_1_1_14_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_14_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_2_1_1_4"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard7.html#T15">GOBOARD7:15</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_2_1_1_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_2_1_1_4"><i><font color="Green">E48</font></i></a>, <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="E3:1_1_1_14_1_2_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="1_1_1_14_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_3"/><i><font color="Green">E46</font></i>: 
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_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="E1:1_1_1_14_1_2_1_1_3_1_1"/>
( ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> ) or ( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_14_1_2"><i><font color="Green">E40</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_14_1_2_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_3_1_1_1"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_3"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_3_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_3_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard7.html#T17">GOBOARD7:17</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_3_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_3_1_1_1"><i><font color="Green">E48</font></i></a>, <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="1_1_1_14_1_2_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_3_1_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_3"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_3_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_14_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_3_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard7.html#T18">GOBOARD7:18</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_3_1_1_2"><i><font color="Green">E48</font></i></a>, <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="1_1_1_14_1_2_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_3_1_1_3"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_3_1_1_3"/><b>then </b><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>13</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></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="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_3"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>13</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>12</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_3_1_1_3"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard7.html#T19">GOBOARD7:19</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_3_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_3_1_1_3"><i><font color="Green">E48</font></i></a>, <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="1_1_1_14_1_2_1_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_14_1_2_1_1_3_1_1_4"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_14_1_2_1_1_3_1_1_4"/><b>then </b><i><font color="Green">E48</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_14_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_3"><i><font color="Green">E46</font></i></a></i>;<br/><a NAME="E3:1_1_1_14_1_2_1_1_3_1_1_4"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E4:1_1_1_14_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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>13</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_14_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_14_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_14_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_14_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_14_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_14_1_2_1_1_3_1_1_4"><i><font color="Green">E47</font></i></a>, <a class="ref" href="goboard7.html#T20">GOBOARD7:20</a></i>;<br/><b>hence </b><a NAME="E5:1_1_1_14_1_2_1_1_3_1_1_4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="txt" href="jordan13.html#E1:1_1_1_14"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_14_1_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_14_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_14_1_2_1_1_3_1_1_4"><i><font color="Green">E48</font></i></a>, <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="E3:1_1_1_14_1_2_1_1_3"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="E15:1_1_1_14_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="E37:1_1_1"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="jordan13.html#E35:1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan13.html#E36:1_1_1"><i><font color="Green">E31</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>7</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 &amp; ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>5</sub></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">c<sub>5</sub></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">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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">c<sub>5</sub></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="1_1_1_15"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_15"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 holds <br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>2</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>2</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>8</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>5</sub></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="E2:1_1_1_15"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>8</sub></font>[0]
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1"><i><font color="Green">E5</font></i></a></i>;<br/><a NAME="E3:1_1_1_15"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>8</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>8</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1_1_15_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_15_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_15_1_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>and </b><br/><a NAME="E2:1_1_1_15_1_1"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:1_1_1_15_1_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E5:1_1_1_15_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1"><i><font color="Green">E25</font></i></a></i>;<br/><a NAME="E6:1_1_1_15_1_1"/><i><font color="Green">E40</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></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">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_15_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:1_1_1_15_1_1"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E8:1_1_1_15_1_1"/><i><font color="Green">E42</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></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="1_1_1_15_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_15_1_1_1"/>
( ( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_15_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_15_1_1"><i><font color="Green">E41</font></i></a>, <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="1_1_1_15_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_15_1_1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_15_1_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
( <font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>10</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_15_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_15_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:1_1_1_15_1_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_15_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_15_1_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_15_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_15_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_15_1_1_1_1"><i><font color="Green">E43</font></i></a>, <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="1_1_1_15_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_15_1_1_1_2"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_15_1_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_15"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_15_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_15_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_15_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_15_1_1"><i><font color="Green">E42</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_15_1_1_1_3"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_15_1_1_1_3"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_15"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_15_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_15_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_15_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_15_1_1"><i><font color="Green">E42</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_15_1_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_15_1_1_1_4"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_15_1_1_1_4"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:1_1_1_15_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></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="E4:1_1_1_15"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>8</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="jordan13.html#E2:1_1_1_15"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_15"><i><font color="Green">E35</font></i></a>);<br/></i><a NAME="E5:1_1_1_15"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:1_1_1_15_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_15"><i><font color="Green">E36</font></i></a></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> = 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="finseq_4.html#T77">FINSEQ_4:77</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>
;<br/><br/>


</div><b>end;</b></div><b>set </b><font color="Maroon">c<sub>8</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/><a NAME="E6:1_1_1_15"/><i><font color="Green">E38</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="E7:1_1_1_15"/><i><font color="Green">E39</font></i>: 
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K2">Values</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="gobrd13.html#T8">GOBRD13:8</a></i>;<br/><a NAME="E8:1_1_1_15"/>
<font color="Maroon">c<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E9:1_1_1_15"/><b>then </b>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="jordan1h.html#K2">Values</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="gobrd13.html#T9">GOBRD13:9</a></i>;<br/><a NAME="E10:1_1_1_15"/><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">c<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="jordan1h.html#K2">Values</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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="E11:1_1_1_15"/><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">c<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_15"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E12:1_1_1_15"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_15"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_15"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E39:1_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E41:1_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>7</sub></font>[<font color="Maroon">c<sub>8</sub></font>]
 <b>and </b><br/><a NAME="E42:1_1_1"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>7</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 <b>from </b><i><a class="ref" href="nat_1.html#S5">NAT_1:sch 5</a>(<a class="txt" href="jordan13.html#E39:1_1_1"><i><font color="Green">E33</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E44:1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E45:1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E46:1_1_1"/><i><font color="Green">E38</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E41:1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<a NAME="E47:1_1_1"/><i><font color="Green">E39</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E48:1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E41:1_1_1"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM8">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="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E50:1_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E51:1_1_1"/><i><font color="Green">E41</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E44:1_1_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E52:1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E45:1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E53:1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E54:1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>10</sub></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/>
<a NAME="E55:1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 2
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></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#NM8">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="jordan13.html#E37:1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E54:1_1_1"><i><font color="Green">E43</font></i></a>, <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">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>;<br/>
<a NAME="E57:1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>9</sub></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">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E52:1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E58:1_1_1"/>
( <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></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="E59:1_1_1"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></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="E60:1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E57:1_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E61:1_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>11</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="rfinseq.html#D2">RFINSEQ:def 2</a></i>;<br/>
<a NAME="E62:1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <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">c<sub>11</sub></font> <a href="rfinseq.html#K1">/^</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="rfinseq.html#K1">/^</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM8">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="E64:1_1_1"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></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">E48</font></i>: <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>
;<br/>
<a NAME="E66:1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></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">E49</font></i>: <font color="Maroon">c<sub>13</sub></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">c<sub>13</sub></font></span>)</span> = 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>13</sub></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">c<sub>11</sub></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">c<sub>11</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E61:1_1_1"><i><font color="Green">E47</font></i></a></i>
;<br/>
<a NAME="E68:1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="jordan8.html#T5">JORDAN8:5</a></i>;<br/>
<a NAME="E69:1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>13</sub></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="E70:1_1_1"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_18"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_18"/><i><font color="Green">E53</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>and </b><br/><a NAME="E2:1_1_1_18"/><i><font color="Green">E54</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>and </b><br/><a NAME="E3:1_1_1_18"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E4:1_1_1_18"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<a NAME="E5:1_1_1_18"/>
<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_18"><i><font color="Green">E53</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E6:1_1_1_18"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>;<br/>
<a NAME="E7:1_1_1_18"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_18"><i><font color="Green">E54</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E8:1_1_1_18"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></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="1_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="E1:1_1_1_18_1"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>14</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>15</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_18"><i><font color="Green">E56</font></i></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="1_1_1_18_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_18_1_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font>;<br/><a NAME="E2:1_1_1_18_1_1"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_18_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:1_1_1_18_1_1"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E61:1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_18"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E4:1_1_1_18_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E5:1_1_1_18_1_1"/>
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:1_1_1_18_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_18"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:1_1_1_18_1_1"/><i><font color="Green">E64</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E8:1_1_1_18_1_1"/>
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:1_1_1_18_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_18"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:1_1_1_18_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_18_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_18_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:1_1_1_18_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></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">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_18_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><a NAME="E12:1_1_1_18_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_18_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_18_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:1_1_1_18_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></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">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_18_1_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><b>hence </b><a NAME="E14:1_1_1_18_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_18"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_18"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_18"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_18_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_18_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_18_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_18_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_18_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_18_1_1"><i><font color="Green">E66</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_18_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_18_1_2"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font>;<br/><a NAME="E2:1_1_1_18_1_2"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_18_1_2"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:1_1_1_18_1_2"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E61:1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_18"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/><a NAME="E4:1_1_1_18_1_2"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green">E28</font></i></a></i>;<br/><a NAME="E5:1_1_1_18_1_2"/>
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:1_1_1_18_1_2"/><b>then </b><i><font color="Green">E63</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_18"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E7:1_1_1_18_1_2"/><i><font color="Green">E64</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E8:1_1_1_18_1_2"/>
0 <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E9:1_1_1_18_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_18"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:1_1_1_18_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_18_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_18_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:1_1_1_18_1_2"/><b>then </b><i><font color="Green">E66</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></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">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_18_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><a NAME="E12:1_1_1_18_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_18_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_18_1_2"><i><font color="Green">E64</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E13:1_1_1_18_1_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></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">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_18_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/><b>hence </b><a NAME="E14:1_1_1_18_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_18"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_18"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_18"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_18_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_18_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_18_1_2"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_18_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_18_1_2"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_18_1_2"><i><font color="Green">E66</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E71:1_1_1"/><i><font color="Green">E53</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_19"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_19"/><i><font color="Green">E54</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E2:1_1_1_19"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E3:1_1_1_19"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E4:1_1_1_19"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<a NAME="E5:1_1_1_19"/><i><font color="Green">E58</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_19"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_19"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:1_1_1_19"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_19"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_19"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:1_1_1_19"/><b>then </b><i><font color="Green">E60</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_19"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_19_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_19_1"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_19_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_19_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><a NAME="E2:1_1_1_19_1_1"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_19"><i><font color="Green">E56</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_19_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_19"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_19"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_19"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_19"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_19"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_19_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_19_1_2"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_19_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E46:1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E65:1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E67:1_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_19"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_19"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_19"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_19"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E72:1_1_1"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> holds <br/><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_20"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_20"/><i><font color="Green">E55</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E2:1_1_1_20"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>14</sub></font>
 <b>and </b><br/><a NAME="E3:1_1_1_20"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E4:1_1_1_20"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<a NAME="E5:1_1_1_20"/><i><font color="Green">E59</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_20"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_20"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E6:1_1_1_20"/>
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_20"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_20"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_1_20"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_20"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_20"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_20"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_20"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_20"><i><font color="Green">E59</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E73:1_1_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></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">c<sub>11</sub></font> holds <br/> <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Olive">b<sub>1</sub></font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_21"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1_1_21"/><i><font color="Green">E56</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></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">c<sub>11</sub></font> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_1_21"/><i><font color="Green">E57</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:1_1_1_21"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E5:1_1_1_21"/><i><font color="Green">E59</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:1_1_1_21"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E7:1_1_1_21"/><i><font color="Green">E61</font></i>: 
( ( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> ) or ( <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> ) or ( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> ) or ( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_21"><i><font color="Green">E56</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/>
<a NAME="E8:1_1_1_21"/><i><font color="Green">E62</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_21"><i><font color="Green">E57</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E9:1_1_1_21"/><i><font color="Green">E63</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_21"><i><font color="Green">E59</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E10:1_1_1_21"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>18</sub></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="1_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="E1:1_1_1_21_1"/>
( ( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> ) or ( <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> ) or ( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> ) or ( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_21"><i><font color="Green">E61</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_21_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_21_1_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_21_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_21"><i><font color="Green">E62</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E3:1_1_1_21_1_1"/>
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_21"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_21"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_21"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_21"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_21"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_21"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_21_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="gobrd13.html#D3">GOBRD13:def 3</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_21_1_1"/>
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_21"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_21_1_1"><i><font color="Green">E66</font></i></a>, <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="1_1_1_21_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_21_1_2"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_21_1_2"/><b>then </b>
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_21"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_21"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_21"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_21"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_21"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_21"><i><font color="Green">E64</font></i></a>, <a class="ref" href="gobrd13.html#D3">GOBRD13:def 3</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_21_1_2"/>
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_21"><i><font color="Green">E62</font></i></a>, <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="1_1_1_21_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_21_1_3"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_21_1_3"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_21"><i><font color="Green">E63</font></i></a>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/><a NAME="E3:1_1_1_21_1_3"/>
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_21"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_21"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_21"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_21"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_21"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_21"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_21_1_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="gobrd13.html#D3">GOBRD13:def 3</a></i>;<br/><b>hence </b><a NAME="E4:1_1_1_21_1_3"/>
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_21"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_21_1_3"><i><font color="Green">E66</font></i></a>, <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="1_1_1_21_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_21_1_4"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_21_1_4"/><b>then </b>
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_21"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_21"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_21"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_21"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_21"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_21"><i><font color="Green">E64</font></i></a>, <a class="ref" href="gobrd13.html#D3">GOBRD13:def 3</a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_21_1_4"/>
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>14</sub></font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_21"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_21"><i><font color="Green">E63</font></i></a>, <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="E74:1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>13</sub></font> is <a href="goboard5.html#V1">s.c.c.</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>14</sub></font> be   <a href="nat_1.html#NM1">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">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_22"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E2:1_1_1_22"/><i><font color="Green">E58</font></i>: 
( ( <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> ) or <font color="Maroon">c<sub>15</sub></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">c<sub>13</sub></font> )
 ;<br/>

<a NAME="E3:1_1_1_22"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></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="E4:1_1_1_22"/><b>then </b><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E5:1_1_1_22"/><i><font color="Green">E61</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></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="E6:1_1_1_22"/><i><font color="Green">E62</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E7:1_1_1_22"/>
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:1_1_1_22"/><b>then </b><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1"/>
( ( <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> ) or ( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>15</sub></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">c<sub>13</sub></font> ) or ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>15</sub></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">c<sub>13</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_22"><i><font color="Green">E58</font></i></a>, <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="1_1_1_22_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>14</sub></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">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:1_1_1_22_1_1"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E4:1_1_1_22_1_1"/><i><font color="Green">E67</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:1_1_1_22_1_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E7:1_1_1_22_1_1"/><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E8:1_1_1_22_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E9:1_1_1_22_1_1"/><i><font color="Green">E71</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E10:1_1_1_22_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E11:1_1_1_22_1_1"/><i><font color="Green">E73</font></i>: 
( ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E68:1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E12:1_1_1_22_1_1"/><i><font color="Green">E74</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_22_1_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E13:1_1_1_22_1_1"/><i><font color="Green">E75</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_22_1_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E14:1_1_1_22_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>15</sub></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">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E15:1_1_1_22_1_1"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>23</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E17:1_1_1_22_1_1"/><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E18:1_1_1_22_1_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E19:1_1_1_22_1_1"/><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E20:1_1_1_22_1_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E21:1_1_1_22_1_1"/><i><font color="Green">E82</font></i>: 
( ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E68:1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E22:1_1_1_22_1_1"/><i><font color="Green">E83</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E17:1_1_1_22_1_1"><i><font color="Green">E78</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E23:1_1_1_22_1_1"/><i><font color="Green">E84</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E19:1_1_1_22_1_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><b>assume </b><a NAME="E24:1_1_1_22_1_1"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/><a NAME="E25:1_1_1_22_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_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="E1:1_1_1_22_1_1_1_1"/>
( ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E11:1_1_1_22_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E21:1_1_1_22_1_1"><i><font color="Green">E82</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1_1_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard7.html#T21">GOBOARD7:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_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="E1:1_1_1_22_1_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T24">GOBOARD7:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_1"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_1"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_1_1_1_3"/>
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_1"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_1_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_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="E1:1_1_1_22_1_1_1_1_2_1_1"/>
( ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_2_1_1_2"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_2"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_2_1_1_3"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_2"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_2_1_1_4"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_2_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_2"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_1_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_3"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_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="E1:1_1_1_22_1_1_1_1_3_1_1"/>
( ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_3"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_3_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_3"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_3_1_1_2"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_3"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_3_1_1_3"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_3_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_3"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_3_1_1_4"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_3_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_3"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_1_1_1_3"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_4"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1_1_1_4"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard7.html#T21">GOBOARD7:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_4_1_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_4"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T24">GOBOARD7:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_4"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_4"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_4_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_4_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_4"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_4"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_4_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_4_1_1_3"/>
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_4_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_4"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_4"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_1_1_1_4"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_5"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_5"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_5_1_1"/>
( ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_5"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_5_1_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_5_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_5_1_1_2"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_5_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_5"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_5_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_5_1_1_3"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_5_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_5"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_5_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_5_1_1_4"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_5_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_5"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_1_1_1_5"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_6"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_6"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1_1_1_6"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard7.html#T22">GOBOARD7:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_6_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_6"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T25">GOBOARD7:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_6_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_6_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_6"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_6_1_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_6_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_6"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_6"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_6_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_6_1_1_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_6_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_6"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_6"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_1_1_1_6"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_7"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_7"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1_1_1_7"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard7.html#T22">GOBOARD7:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_7_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> or <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_7"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T25">GOBOARD7:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_7_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_7_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_7_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_7"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_7"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_7_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_7_1_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_7_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_7"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_7"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_7_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_7_1_1_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_7_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_7"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_7"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_1_1_1_7"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_8"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_8"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_8_1_1"/>
( ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_8"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_8_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_8_1_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_8_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_8"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_8_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_8_1_1_2"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_8_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_8"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_8_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_8_1_1_3"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_8_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_8"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_8_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_8_1_1_4"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_8_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_8"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_1_1_1_8"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_9"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_9"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_9_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_9_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_9_1_1"/>
( ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_9"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_9_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_9_1_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_9_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_9"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_9_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_9_1_1_2"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_9_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_9"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_9_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_9_1_1_3"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_9_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_9"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_9_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_9_1_1_4"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_9_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_9"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_1_1_1_9"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_10"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_10"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1_1_1_10"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard7.html#T22">GOBOARD7:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_10_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_10_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_10"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T25">GOBOARD7:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_10_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_10_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_10_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_10"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_10"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_10_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_10_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_10_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_10"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_10"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_10_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_10_1_1_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_10_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_10"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_10"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_1_1_1_10"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_11"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_11"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1_1_1_11"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard7.html#T22">GOBOARD7:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_11_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_11_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_11"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T25">GOBOARD7:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_11_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_11_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_11_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_11"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_11"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_11_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_11_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_11_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_11"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_11"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_11_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_11_1_1_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_11_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_11"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_11"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_1_1_1_11"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_12"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_12"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_12_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_12_1_1"/>
( ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_12"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_12_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_12_1_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_12_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_12"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_12_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_12_1_1_2"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_12_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_12"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_12_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_12_1_1_3"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_12_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_12"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_12_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_12_1_1_4"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_12_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_12"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_1_1_1_12"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_13"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_13"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1_1_1_13"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard7.html#T21">GOBOARD7:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_13_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_13_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_13_1_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_13"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T24">GOBOARD7:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_13_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_13_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_13_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_13"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_13"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_13_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_13_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_13_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_13"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_13"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_13_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_13_1_1_3"/>
<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_13_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_13"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_13"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_1_1_1_13"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_14"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_14"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_14_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_14_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_14_1_1"/>
( ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_14"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_14_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_14_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_14_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_14"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_14_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_14_1_1_2"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_14_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_14"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_14_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_14_1_1_3"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_14_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_14"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_14_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_14_1_1_4"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_14_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_14"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_1_1_1_14"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_15"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_15"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_15_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_15_1_1"/>
( ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_15"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_15_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_15_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_15_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_15"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_15_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_15_1_1_2"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_15_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_15"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_15_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_15_1_1_3"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_15_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_15"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_15_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_15_1_1_4"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_15_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_15"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_1_1_1_15"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_16"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_16"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_1_1_1_16"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard7.html#T21">GOBOARD7:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_16_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_16_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_1_1_1_16_1_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_22_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_1"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_1"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_16"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T24">GOBOARD7:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_16_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_16_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_16_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_16"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_16"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_16_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_16_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_16_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_22_1_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_16"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_16"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_1_1_1_16_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_1_1_1_16_1_1_3"/>
<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_1_1_1_16_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_1_1_1_16"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_1_1_1_16"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_1_1_1_16"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E27:1_1_1_22_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_2"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">c<sub>15</sub></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">c<sub>13</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_2"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></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="E3:1_1_1_22_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3"/><i><font color="Green">E64</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>15</sub></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">c<sub>13</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>14</sub></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">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E3:1_1_1_22_1_3"/><b>then </b><i><font color="Green">E66</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><a NAME="E4:1_1_1_22_1_3"/><i><font color="Green">E67</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:1_1_1_22_1_3"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E6:1_1_1_22_1_3"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>16</sub></font>, <font color="Maroon">c<sub>17</sub></font>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E8:1_1_1_22_1_3"/><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E9:1_1_1_22_1_3"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E10:1_1_1_22_1_3"/><i><font color="Green">E72</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E11:1_1_1_22_1_3"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E12:1_1_1_22_1_3"/><i><font color="Green">E74</font></i>: 
( ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E68:1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green">E65</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E13:1_1_1_22_1_3"/><i><font color="Green">E75</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>17</sub></font> &amp; <font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_22_1_3"><i><font color="Green">E70</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E14:1_1_1_22_1_3"/><i><font color="Green">E76</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>19</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_22_1_3"><i><font color="Green">E72</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E15:1_1_1_22_1_3"/><i><font color="Green">E77</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>20</sub></font>, <font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>23</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E17:1_1_1_22_1_3"/><i><font color="Green">E78</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E18:1_1_1_22_1_3"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>20</sub></font>,<font color="Maroon">c<sub>21</sub></font>
 <b>and </b><br/><a NAME="E19:1_1_1_22_1_3"/><i><font color="Green">E80</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E20:1_1_1_22_1_3"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>22</sub></font>,<font color="Maroon">c<sub>23</sub></font>
 <b>and </b><br/><a NAME="E21:1_1_1_22_1_3"/><i><font color="Green">E82</font></i>: 
( ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E68:1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E22:1_1_1_22_1_3"/><i><font color="Green">E83</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>20</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E17:1_1_1_22_1_3"><i><font color="Green">E78</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E23:1_1_1_22_1_3"/><i><font color="Green">E84</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>23</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E19:1_1_1_22_1_3"><i><font color="Green">E80</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><b>assume </b><a NAME="E24:1_1_1_22_1_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/><a NAME="E25:1_1_1_22_1_3"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_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="E1:1_1_1_22_1_3_1_1"/>
( ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E12:1_1_1_22_1_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E21:1_1_1_22_1_3"><i><font color="Green">E82</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_1"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3_1_1_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard7.html#T21">GOBOARD7:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_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="E1:1_1_1_22_1_3_1_1_1_1_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T24">GOBOARD7:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_1"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E72:1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_1"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_1_1_1_3"/>
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_1"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_1"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_3_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_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="E1:1_1_1_22_1_3_1_1_2_1_1"/>
( ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_2"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_2_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_2_1_1_2"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_2"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_2_1_1_3"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E72:1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_2"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_2_1_1_4"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_2_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_2"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_3_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_3"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_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="E1:1_1_1_22_1_3_1_1_3_1_1"/>
( ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_3"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_3_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E72:1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_3"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_3_1_1_2"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_3_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_3"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_3_1_1_3"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_3_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_3"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_3_1_1_4"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_3_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_3"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_3_1_1_3"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_4"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3_1_1_4"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard7.html#T21">GOBOARD7:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_4_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_4_1_1"/>
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_4"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T24">GOBOARD7:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_4_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_4_1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_4"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_4"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_4_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_4_1_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_4_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_4"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_4"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_4_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_4_1_1_3"/>
<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_4_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_4"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_4"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_3_1_1_4"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_5"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_5"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_5_1_1"/>
( ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_5"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_5_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_5_1_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_5_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_5_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_5_1_1_2"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_5_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E72:1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_5"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_5_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_5_1_1_3"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_5_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_5"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_5_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_5_1_1_4"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_5_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_5"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_3_1_1_5"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_6"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_6"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3_1_1_6"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard7.html#T22">GOBOARD7:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_6_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_6"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T25">GOBOARD7:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_6_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_6_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_6_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_6"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_6_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_6_1_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_6_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E72:1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_6"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_6"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_6_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_6_1_1_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_6_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_6"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_6"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_3_1_1_6"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_7"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_7"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3_1_1_7"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard7.html#T22">GOBOARD7:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_7_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_7_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_7_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> or <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_7"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T25">GOBOARD7:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_7_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_7_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_7_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E72:1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_7"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_7"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_7_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_7_1_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_7_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_7"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_7"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_7_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_7_1_1_3"/>
<font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_7_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_7"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_7"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_3_1_1_7"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_8"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_8"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_8_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_8_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_8_1_1"/>
( ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_8"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_8_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_8_1_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_8_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E72:1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_8"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_8_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_8_1_1_2"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_8_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_8"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_8_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_8_1_1_3"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_8_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_8"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_8_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_8_1_1_4"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_8_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_8"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_3_1_1_8"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_9"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_9"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_9_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_9_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_9_1_1"/>
( ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_9"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_9_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_9_1_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_9_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_9"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_9_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_9_1_1_2"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_9_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_9"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_9_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_9_1_1_3"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_9_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_9"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_9_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_9_1_1_4"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_9_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E72:1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_9"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_3_1_1_9"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_10"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_10"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3_1_1_10"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard7.html#T22">GOBOARD7:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_10_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_10_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_10_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_10"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T25">GOBOARD7:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_10_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_10_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_10_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_10"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_10"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_10_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_10_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_10_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_10"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_10"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_10_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_10_1_1_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_10_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_10"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_10"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_3_1_1_10"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_11"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_11"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3_1_1_11"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard7.html#T22">GOBOARD7:22</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_11_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_11_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_11_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_11"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T25">GOBOARD7:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_11_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_11_1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_11_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_11"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_11"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_11_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_11_1_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_11_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_11"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_11"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_11_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_11_1_1_3"/>
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_11_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E72:1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_11"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_11"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_3_1_1_11"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_12"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_12"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_12_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_12_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_12_1_1"/>
( ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) or ( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_12"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_12_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_12_1_1_1"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_12_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_12"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_12_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_12_1_1_2"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_12_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_12"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_12_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_12_1_1_3"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_12_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E72:1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_12"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_12_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_12_1_1_4"/>
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>17</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_12_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_12"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_3_1_1_12"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_13"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_13"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3_1_1_13"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard7.html#T21">GOBOARD7:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_13_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_13_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_13_1_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_13"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T24">GOBOARD7:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_13_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_13_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_13_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_13"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_13"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_13_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_13_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_13_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_13"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_13"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_13_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_13_1_1_3"/>
<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_13_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_13"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_13"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_3_1_1_13"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_14"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_14"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_14_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_14_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_14_1_1"/>
( ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_14"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_14_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_14_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_14_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_14"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_14_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_14_1_1_2"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_14_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_14"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_14_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_14_1_1_3"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_14_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_14"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_14_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_14_1_1_4"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_14_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E72:1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_14"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_3_1_1_14"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_15"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_15"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_15_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_15_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_15_1_1"/>
( ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) or ( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_15"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T23">GOBOARD7:23</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_15_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_15_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_15_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_15"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_15_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_15_1_1_2"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_15_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E72:1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_15"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_15_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_15_1_1_3"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_15_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_15"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_15_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_15_1_1_4"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_15_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_22"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22_1_3"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_15"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_22_1_3_1_1_15"/>
contradiction
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_16"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_16"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_22_1_3_1_1_16"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="goboard7.html#T21">GOBOARD7:21</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_16_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_16_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_22_1_3_1_1_16_1_1"/>
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> or <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 or <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_22_1_3"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_22_1_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_22_1_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_22_1_3"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_22_1_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_22_1_3"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_22_1_3"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_16"><i><font color="Green">E86</font></i></a>, <a class="ref" href="goboard7.html#T24">GOBOARD7:24</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_16_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_16_1_1_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_16_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E71:1_1_1"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_22_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_16"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_16"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_16_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_16_1_1_2"/>
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_16_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E70:1_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_22"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_22_1_3"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_22_1_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_22_1_3"><i><font color="Green">E79</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_16"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_16"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_22_1_3_1_1_16_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_22_1_3_1_1_16_1_1_3"/>
<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:1_1_1_22_1_3_1_1_16_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E72:1_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_22"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_22_1_3"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_22_1_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_22_1_3_1_1_16"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_22_1_3_1_1_16"><i><font color="Green">E87</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:1_1_1_22_1_3_1_1_16"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E27:1_1_1_22_1_3"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E75:1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font> <a href="binarith.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">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E57:1_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E61:1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E76:1_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></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">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="binarith.html#T50">BINARITH:50</a></i>;<br/>
<a NAME="E77:1_1_1"/><b>then </b><i><font color="Green">E58</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">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>9</sub></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">c<sub>13</sub></font>
 
;<br/>
<a NAME="E78:1_1_1"/>
not <font color="Maroon">c<sub>13</sub></font> is <a href="funct_1.html#V3">constant</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_23"><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="E1:1_1_1_23"/><i><font color="Green">E59</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/>

<b>thus </b><a NAME="E2:1_1_1_23"/><i><font color="Green">E60</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E76:1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>

<a NAME="E3:1_1_1_23"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></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="txt" href="jordan13.html#E69:1_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_23"><i><font color="Green">E59</font></i></a>, <a class="ref" href="goboard7.html#T31">GOBOARD7:31</a></i>;<br/>
<a NAME="E4:1_1_1_23"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></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="txt" href="jordan13.html#E1:1_1_1_23"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E5:1_1_1_23"/>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> 2
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_23"><i><font color="Green">E60</font></i></a>, <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">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></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="txt" href="jordan13.html#E46:1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E65:1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E67:1_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E68:1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E74:1_1_1"><i><font color="Green">E56</font></i></a>, <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/>
<a NAME="E80:1_1_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_24"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_24"/><i><font color="Green">E60</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font>
 <b>and </b><br/><a NAME="E2:1_1_1_24"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<a NAME="E3:1_1_1_24"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_24"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E4:1_1_1_24"/>
<font color="Maroon">c<sub>15</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_24"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E5:1_1_1_24"/><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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/>
<a NAME="E6:1_1_1_24"/><b>then </b>
<font color="Maroon">c<sub>15</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_24"><i><font color="Green">E60</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>hence </b><a NAME="E7:1_1_1_24"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_24"><i><font color="Green">E62</font></i></a>, <a class="ref" href="finseq_4.html#T85">FINSEQ_4:85</a></i>;<br/>


</div><b>end;</b></div>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></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="E82:1_1_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>15</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_25"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></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="E1:1_1_1_25"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:1_1_1_25"/><i><font color="Green">E62</font></i>: 
not <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></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><a class="txt" href="jordan13.html#E1:1_1_1_25"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E4:1_1_1_25"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_25"><i><font color="Green">E62</font></i></a>, <a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:1_1_1_25"/><i><font color="Green">E63</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></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">c<sub>14</sub></font> )
 <b>and </b><br/><a NAME="E7:1_1_1_25"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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="E8:1_1_1_25"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>18</sub></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">c<sub>14</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_25"><i><font color="Green">E63</font></i></a>, <a class="ref" href="goboard2.html#T3">GOBOARD2:3</a></i>;<br/>
<a NAME="E9:1_1_1_25"/><b>then </b><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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="E10:1_1_1_25"/><i><font color="Green">E66</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_25"><i><font color="Green">E63</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E11:1_1_1_25"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E12:1_1_1_25"/><b>then </b><i><font color="Green">E67</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_25"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E13:1_1_1_25"/><b>then </b><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E61:1_1_1"><i><font color="Green">E47</font></i></a></i>;<br/>
<a NAME="E14:1_1_1_25"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E61:1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_25"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_25"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_25"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_25"><i><font color="Green">E67</font></i></a>, <a class="ref" href="topreal1.html#D5">TOPREAL1:def 5</a></i>;<br/>
<a NAME="E15:1_1_1_25"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_25"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_25"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T30">GOBRD13:30</a></i>;<br/>
<a NAME="E16:1_1_1_25"/><b>then </b>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E17:1_1_1_25"/><b>then </b>
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_25"><i><font color="Green">E61</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E18:1_1_1_25"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_25"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_25"><i><font color="Green">E68</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E83:1_1_1"/><i><font color="Green">E61</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="E84:1_1_1"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></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="E85:1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></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="E86:1_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></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="E87:1_1_1"/><i><font color="Green">E63</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T75">JORDAN3:75</a></i>;<br/>
<a NAME="E88:1_1_1"/><i><font color="Green">E64</font></i>: 
(  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  &amp;  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></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/>
<a NAME="E89:1_1_1"/><i><font color="Green">E65</font></i>: 
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>11</sub></font> holds <br/>for <font color="Olive">b<sub>2</sub></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">b<sub>2</sub></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">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K3">`</a>  holds <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></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">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> &amp; not <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_26"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></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="E1:1_1_1_26"/><i><font color="Green">E66</font></i>: 
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>17</sub></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="E2:1_1_1_26"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>17</sub></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">c<sub>16</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E3:1_1_1_26"/><i><font color="Green">E68</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></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">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E4:1_1_1_26"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E5:1_1_1_26"/><i><font color="Green">E70</font></i>: 
not <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>19</sub></font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>16</sub></font></span>)</span>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>16</sub></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="E7:1_1_1_26"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a>  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T36">XBOOLE_1:36</a></i>;<br/>
<a NAME="E8:1_1_1_26"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>16</sub></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="E9:1_1_1_26"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_26_1"><b>proof </b></a><div class="add">

<a NAME="E1:1_1_1_26_1"/>
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_26"><i><font color="Green">E68</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>21</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_1_26_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></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">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></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="jordan13.html#E3:1_1_1_26_1"><i><font color="Green">E74</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_26_1_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_26_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:1_1_1_26_1_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_1_26_1_1_1"/><i><font color="Green">E75</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>and </b><br/><a NAME="E4:1_1_1_26_1_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>24</sub></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">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E5:1_1_1_26_1_1_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_26"><i><font color="Green">E66</font></i></a>, <a class="ref" href="sppol_2.html#T13">SPPOL_2:13</a></i>;<br/><a NAME="E6:1_1_1_26_1_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>24</sub></font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>24</sub></font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_26_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_26_1_1_1"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_26_1_1_1"><i><font color="Green">E77</font></i></a>, <a class="ref" href="gobrd13.html#T30">GOBRD13:30</a></i>;<br/><a NAME="E7:1_1_1_26_1_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>24</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E8:1_1_1_26_1_1_1"/>
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>24</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_26_1_1_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_26_1_1_1"><i><font color="Green">E76</font></i></a></i>;<br/><b>hence </b><a NAME="E9:1_1_1_26_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_26_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_26_1_1_1"><i><font color="Green">E78</font></i></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="E2:1_1_1_26_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_26_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>thus </b><a NAME="E3:1_1_1_26_1_1"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_26_1"><i><font color="Green">E74</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:1_1_1_26_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></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="E10:1_1_1_26"/><i><font color="Green">E74</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_26"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jordan9.html#T12">JORDAN9:12</a></i>;<br/>
<a NAME="E11:1_1_1_26"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_26"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jordan9.html#T17">JORDAN9:17</a></i>;<br/>
<a NAME="E12:1_1_1_26"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_26"><i><font color="Green">E66</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E13:1_1_1_26"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>16</sub></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="E14:1_1_1_26"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E15:1_1_1_26"/><b>then </b><i><font color="Green">E76</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_26"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_26"><i><font color="Green">E75</font></i></a>, <a class="ref" href="xboole_1.html#T19">XBOOLE_1:19</a></i>;<br/>
<a NAME="E16:1_1_1_26"/>
<font color="Maroon">c<sub>20</sub></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#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_26"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_26"><i><font color="Green">E71</font></i></a>, <a class="ref" href="jordan9.html#T13">JORDAN9:13</a></i>;<br/>
<a NAME="E17:1_1_1_26"/><b>then </b><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>20</sub></font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_26"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_26"><i><font color="Green">E76</font></i></a>, <a class="ref" href="connsp_1.html#T19">CONNSP_1:19</a></i>;<br/>
<a NAME="E18:1_1_1_26"/>
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_26"><i><font color="Green">E68</font></i></a>, <a class="ref" href="jordan9.html#T10">JORDAN9:10</a></i>;<br/>
<a NAME="E19:1_1_1_26"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>20</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_26"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E20:1_1_1_26"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_26"><i><font color="Green">E69</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E21:1_1_1_26"/>
<font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_26"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E22:1_1_1_26"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_26"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_26"><i><font color="Green">E77</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_26"><i><font color="Green">E78</font></i></a>, <a class="ref" href="goboard9.html#T6">GOBOARD9:6</a></i>;<br/>
<b>hence </b><a NAME="E23:1_1_1_26"/>
<font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>17</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_26"><i><font color="Green">E73</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E90:1_1_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>14</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_27"><b>proof </b></a><div class="add">

<a NAME="E1:1_1_1_27"/>
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E57:1_1_1"><i><font color="Green">E45</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>16</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:1_1_1_27"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></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">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></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="jordan13.html#E3:1_1_1_27"><i><font color="Green">E67</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_27_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>17</sub></font>;<br/><b>thus </b><a NAME="E1:1_1_1_27_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_27"><i><font color="Green">E67</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></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">c<sub>20</sub></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="E4:1_1_1_27_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E5:1_1_1_27_1"/><i><font color="Green">E69</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>20</sub></font> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="txt" href="jordan13.html#E82:1_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E86:1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_27"><i><font color="Green">E67</font></i></a>, <a class="ref" href="goboard6.html#T8">GOBOARD6:8</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></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="E7:1_1_1_27_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_27_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="tbsp_1.html#T16">TBSP_1:16</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>22</sub></font> =  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>19</sub></font>,<font color="Maroon">c<sub>21</sub></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="txt" href="jordan13.html#E83:1_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_27_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="tbsp_1.html#T16">TBSP_1:16</a>, <a class="ref" href="topmetr.html#T16">TOPMETR:16</a></i>;<br/><a NAME="E9:1_1_1_27_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>22</sub></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="E10:1_1_1_27_1"/>
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>14</sub></font>,1,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">c<sub>14</sub></font>,1
 <b>by </b><i><a class="txt" href="jordan13.html#E68:1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan13.html#E77:1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="gobrd13.html#T34">GOBRD13:34</a></i>;<br/><a NAME="E11:1_1_1_27_1"/><b>then </b><i><font color="Green">E72</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>14</sub></font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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#K5">left_cell</a> <font color="Maroon">c<sub>14</sub></font>,1</span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/><a NAME="E12:1_1_1_27_1"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K5">left_cell</a> <font color="Maroon">c<sub>14</sub></font>,1</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E77:1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard9.html#T24">GOBOARD9:24</a></i>;<br/><a NAME="E13:1_1_1_27_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>14</sub></font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E11:1_1_1_27_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E14:1_1_1_27_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E77:1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="gobrd13.html#T33">GOBRD13:33</a></i>;<br/><a NAME="E15:1_1_1_27_1"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E16:1_1_1_27_1"/><i><font color="Green">E74</font></i>: 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E57:1_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E73:1_1_1"><i><font color="Green">E55</font></i></a></i>;<br/><a NAME="E17:1_1_1_27_1"/>
<font color="Maroon">c<sub>22</sub></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="E18:1_1_1_27_1"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>9</sub></font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_27"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_27_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_27_1"><i><font color="Green">E74</font></i></a>, <a class="ref" href="tops_1.html#T39">TOPS_1:39</a></i>;<br/><a NAME="E19:1_1_1_27_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_27_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="tbsp_1.html#T16">TBSP_1:16</a></i>;<br/><a NAME="E20:1_1_1_27_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E88:1_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_27_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_27_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_27_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_27_1"><i><font color="Green">E75</font></i></a>, <a class="ref" href="goboard9.html#T6">GOBOARD9:6</a></i>;<br/><b>hence </b><a NAME="E21:1_1_1_27_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E19:1_1_1_27_1"><i><font color="Green">E76</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:1_1_1_27"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>14</sub></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="E91:1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:1_1_1_28"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E2:1_1_1_28"/><b>then </b><i><font color="Green">E67</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/>
<a NAME="E3:1_1_1_28"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1 holds <br/> not <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:1_1_1_28_1"/><i><font color="Green">E69</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E2:1_1_1_28_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1
 ;<br/>

<a NAME="E3:1_1_1_28_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font>;<br/>
<b>assume </b><a NAME="E4:1_1_1_28_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:1_1_1_28_1"/><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E7:1_1_1_28_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>18</sub></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">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E8:1_1_1_28_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan9.html#T9">JORDAN9:9</a></i>;<br/>
<a NAME="E9:1_1_1_28_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:1_1_1_28_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1_28_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E11:1_1_1_28_1"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E12:1_1_1_28_1"/><b>then </b><i><font color="Green">E78</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E13:1_1_1_28_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_1"><i><font color="Green">E73</font></i></a></i>;<br/>
<a NAME="E14:1_1_1_28_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E15:1_1_1_28_1"/><i><font color="Green">E81</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1"><i><font color="Green">E2</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E16:1_1_1_28_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan1h.html#K2">Values</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="jordan9.html#T8">JORDAN9:8</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_28_1_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font> or <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_28_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_1"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jordan9.html#T25">JORDAN9:25</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_28_1_1_1"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_1_1_1"/>
<font color="Maroon">c<sub>16</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_28_1"><i><font color="Green">E76</font></i></a></i>;<br/><a NAME="E3:1_1_1_28_1_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>16</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:1_1_1_28_1_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>16</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_1"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_1"><i><font color="Green">E75</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_28_1"><i><font color="Green">E76</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_1"><i><font color="Green">E71</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>18</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_28_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1_1_1"><i><font color="Green">E83</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>18</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>18</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>
;<br/><b>hence </b><a NAME="E6:1_1_1_28_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E12:1_1_1_28_1"><i><font color="Green">E78</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_1"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_1_1_1"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_1_1_1"><i><font color="Green">E85</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_28_1_1_2"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></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="1_1_1_28_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_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="E1:1_1_1_28_1_1_2_1_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> or <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E13:1_1_1_28_1"><i><font color="Green">E79</font></i></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="1_1_1_28_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_28_1_1_2_1_1_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_1_1_2_1_1_1"/>
<font color="Maroon">c<sub>16</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_28_1"><i><font color="Green">E77</font></i></a></i>;<br/><a NAME="E3:1_1_1_28_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E85</font></i>: 
<font color="Maroon">c<sub>16</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E4:1_1_1_28_1_1_2_1_1_1"/><i><font color="Green">E86</font></i>: 
<font color="Maroon">c<sub>16</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_28_1"><i><font color="Green">E77</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_1"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_28_1"><i><font color="Green">E77</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_1"><i><font color="Green">E71</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E46:1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1_1_2_1_1_1"><i><font color="Green">E84</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>
;<br/><b>hence </b><a NAME="E6:1_1_1_28_1_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E52:1_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_1_1_2_1_1_1"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_1_1_2_1_1_1"><i><font color="Green">E86</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_28_1_1_2_1_1_2"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/><a NAME="E2:1_1_1_28_1_1_2_1_1_2"/><i><font color="Green">E85</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></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="E3:1_1_1_28_1_1_2_1_1_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_28_1"><i><font color="Green">E76</font></i></a>, <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="E4:1_1_1_28_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E87</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>18</sub></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="E5:1_1_1_28_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>16</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E6:1_1_1_28_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E88</font></i>: 
<font color="Maroon">c<sub>16</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green">E69</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E7:1_1_1_28_1_1_2_1_1_2"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">c<sub>16</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_1_1_2_1_1_2"><i><font color="Green">E86</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_1_1_2_1_1_2"><i><font color="Green">E87</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>16</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_1"><i><font color="Green">E71</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1_1_2"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1_1_2_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_1_1_2_1_1_2"><i><font color="Green">E85</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>
;<br/><b>hence </b><a NAME="E9:1_1_1_28_1_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_1_1_2_1_1_2"><i><font color="Green">E84</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_1_1_2_1_1_2"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_1_1_2_1_1_2"><i><font color="Green">E88</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_1_1_2_1_1_2"><i><font color="Green">E89</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:1_1_1_28_1_1_2"/>
contradiction
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E4:1_1_1_28"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>16</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:1_1_1_28_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font>
 ;<br/>

<b>hence </b><span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="binarith.html#K5">-'</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"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</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/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1

;<br/><br/>


</div><b>end;</b></div>
<a NAME="E5:1_1_1_28"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_3"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>16</sub></font> =  <a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font>;<br/>
<a NAME="E1:1_1_1_28_3"/><i><font color="Green">E70</font></i>: 
 <a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E68:1_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="jordan9.html#T7">JORDAN9:7</a></i>;<br/>
<a NAME="E2:1_1_1_28_3"/><i><font color="Green">E71</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">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E76:1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/>
<a NAME="E3:1_1_1_28_3"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.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> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/>
<a NAME="E4:1_1_1_28_3"/><b>then </b><i><font color="Green">E72</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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>
<a NAME="E5:1_1_1_28_3"/>
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">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E6:1_1_1_28_3"/><b>then </b><i><font color="Green">E73</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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>14</sub></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">c<sub>17</sub></font> = <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> = <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2;<br/>
<b>set </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>;<br/>
<a NAME="E7:1_1_1_28_3"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E46:1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E67:1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finseq_5.html#T68">FINSEQ_5:68</a></i>;<br/>
<a NAME="E8:1_1_1_28_3"/><i><font color="Green">E75</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E9:1_1_1_28_3"/><b>then </b><i><font color="Green">E76</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28"><i><font color="Green">E67</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E10:1_1_1_28_3"/>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E76:1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/>
<a NAME="E11:1_1_1_28_3"/><b>then </b><i><font color="Green">E77</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>
<a NAME="E12:1_1_1_28_3"/>
<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">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E76:1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<a NAME="E13:1_1_1_28_3"/><b>then </b><i><font color="Green">E78</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">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
 
;<br/>
<a NAME="E14:1_1_1_28_3"/><i><font color="Green">E79</font></i>: 
1 <a href="real_1.html#K5">-</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="real_1.html#K5">-</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28"><i><font color="Green">E67</font></i></a>, <a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/>
<i><font color="Green">E80</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E11:1_1_1_28_3"><i><font color="Green">E77</font></i></a>, <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">c<sub>14</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E13:1_1_1_28_3"><i><font color="Green">E78</font></i></a>, <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">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></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">c<sub>9</sub></font></span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E61:1_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3"><i><font color="Green">E79</font></i></a>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K5">-</a> 1

;<br/>
<a NAME="E16:1_1_1_28_3"/><b>then </b><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
;<br/>
<a NAME="E17:1_1_1_28_3"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/>
<a NAME="E18:1_1_1_28_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E19:1_1_1_28_3"/><b>then </b><i><font color="Green">E83</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E11:1_1_1_28_3"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E20:1_1_1_28_3"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/>
<a NAME="E21:1_1_1_28_3"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E11:1_1_1_28_3"><i><font color="Green">E77</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E22:1_1_1_28_3"/>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E77:1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="sppol_1.html#T5">SPPOL_1:5</a></i>;<br/>
<a NAME="E23:1_1_1_28_3"/><b>then </b><i><font color="Green">E85</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_28"><i><font color="Green">E69</font></i></a></i>;<br/>
<b>then </b><i><font color="Green">E86</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 = 
<font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E21:1_1_1_28_3"><i><font color="Green">E84</font></i></a>, <a class="ref" href="finseq_5.html#T69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E21:1_1_1_28_3"><i><font color="Green">E84</font></i></a>, <a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>
;<br/>
<a NAME="E25:1_1_1_28_3"/><i><font color="Green">E87</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E67:1_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green">E80</font></i></a>, <a class="ref" href="finseq_5.html#T68">FINSEQ_5:68</a></i>;<br/>
<a NAME="E26:1_1_1_28_3"/><i><font color="Green">E88</font></i>: 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a></i>;<br/>
<a NAME="E27:1_1_1_28_3"/><i><font color="Green">E89</font></i>: 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a></i>;<br/>
<a NAME="E28:1_1_1_28_3"/>
not <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_28"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a></i>;<br/>
<a NAME="E29:1_1_1_28_3"/><b>then </b><i><font color="Green">E90</font></i>: 
not <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="sppol_2.html#T22">SPPOL_2:22</a></i>;<br/>
<a NAME="E30:1_1_1_28_3"/><i><font color="Green">E91</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28"><i><font color="Green">E69</font></i></a></i>;<br/>
<a NAME="E31:1_1_1_28_3"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E32:1_1_1_28_3"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<a NAME="E33:1_1_1_28_3"/><b>then </b><i><font color="Green">E93</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E34:1_1_1_28_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E11:1_1_1_28_3"><i><font color="Green">E77</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E35:1_1_1_28_3"/><b>then </b>
<font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E36:1_1_1_28_3"/><b>then </b><i><font color="Green">E94</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font>, <font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>24</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E38:1_1_1_28_3"/><i><font color="Green">E95</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E39:1_1_1_28_3"/><i><font color="Green">E96</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E40:1_1_1_28_3"/><i><font color="Green">E97</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E41:1_1_1_28_3"/><i><font color="Green">E98</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>
 <b>and </b><br/><a NAME="E42:1_1_1_28_3"/><i><font color="Green">E99</font></i>: 
( ( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> ) or ( <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> ) or ( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> ) or ( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3"><i><font color="Green">E71</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/>
<a NAME="E43:1_1_1_28_3"/><i><font color="Green">E100</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>22</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:1_1_1_28_3_3"/>
( ( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> ) or ( <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> ) or ( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> ) or ( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1_28_3"><i><font color="Green">E99</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_1"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_1"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>26</sub></font>, <font color="Maroon">c<sub>27</sub></font>, <font color="Maroon">c<sub>28</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_1_28_3_3_1"/><i><font color="Green">E102</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:1_1_1_28_3_3_1"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E5:1_1_1_28_3_3_1"/><i><font color="Green">E104</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>28</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:1_1_1_28_3_3_1"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>28</sub></font>
 <b>and </b><br/><a NAME="E7:1_1_1_28_3_3_1"/><i><font color="Green">E106</font></i>: 
( ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E8:1_1_1_28_3_3_1"/><i><font color="Green">E107</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>26</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_1"><i><font color="Green">E102</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><b>thus </b><a NAME="E9:1_1_1_28_3_3_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_3_3_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_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="E1:1_1_1_28_3_3_1_1_1"/>
( ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3_3_1"><i><font color="Green">E106</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_1_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_1_1_1_1"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_1_1_1_1"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E4:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E5:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E57:1_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E7:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>26</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><i><font color="Green">E114</font></i>: <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E111</font></i></a></i>
;<br/><a NAME="E9:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a></i>;<br/> <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan13.html#E41:1_1_1_28_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>
<br/>.= 
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_1"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E115</font></i></a>, <a class="ref" href="gobrd13.html#T36">GOBRD13:36</a></i>
;<br/><a NAME="E11:1_1_1_28_3_3_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R1">turns_right</a> <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E33:1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1_28_3"><i><font color="Green">E89</font></i></a></i>;<br/><a NAME="E12:1_1_1_28_3_3_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="gobrd13.html#R1">turns_right</a> <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E57:1_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_28_3"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E112</font></i></a>, <a class="ref" href="gobrd13.html#T44">GOBRD13:44</a></i>;<br/><a NAME="E13:1_1_1_28_3_3_1_1_1_1"/><b>then </b><i><font color="Green">E116</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>22</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>22</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_28_3"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_1"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E113</font></i></a>, <a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/><a NAME="E14:1_1_1_28_3_3_1_1_1_1"/><b>then </b><i><font color="Green">E117</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>29</sub></font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>14</sub></font></span>)</span></span>)</span>;<br/><a NAME="E15:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jordan9.html#T29">JORDAN9:29</a></i>;<br/><a NAME="E16:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green">E119</font></i>: 
( <font color="Maroon">c<sub>22</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E43:1_1_1_28_3"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E108</font></i></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="E17:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green">E120</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E76:1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E18:1_1_1_28_3_3_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green">E121</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> = 
<font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E17:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E120</font></i></a>, <a class="ref" href="finseq_5.html#T69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E30:1_1_1_28_3"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E120</font></i></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">c<sub>14</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E46:1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E65:1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E67:1_1_1"><i><font color="Green">E49</font></i></a>, <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">c<sub>14</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> <font color="Maroon">c<sub>14</sub></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">c<sub>14</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"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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="E21:1_1_1_28_3_3_1_1_1_1"/><b>then </b>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E116</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E121</font></i></a>, <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>;<br/><a NAME="E22:1_1_1_28_3_3_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E43:1_1_1_28_3"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_1"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3_3_1"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E117</font></i></a>, <a class="ref" href="jordan9.html#T22">JORDAN9:22</a></i>;<br/><a NAME="E23:1_1_1_28_3_3_1_1_1_1"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>14</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green">E90</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E24:1_1_1_28_3_3_1_1_1_1"/><i><font color="Green">E123</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">c<sub>14</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E87:1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="sppol_2.html#T22">SPPOL_2:22</a></i>;<br/><a NAME="E25:1_1_1_28_3_3_1_1_1_1"/>
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D1">GOBOARD9:def 1</a></i>;<br/><b>hence </b><a NAME="E26:1_1_1_28_3_3_1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E89:1_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan13.html#E23:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3_3_1_1_1_1"><i><font color="Green">E123</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_1_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_1_1_1_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_1_1_1_2"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_1"><i><font color="Green">E101</font></i></a>, <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">c<sub>11</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan13.html#E41:1_1_1_28_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_1_1_1_2"><i><font color="Green">E109</font></i></a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>
<br/>.= 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_1"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_1"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_1_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>
;<br/><b>hence </b><a NAME="E5:1_1_1_28_3_3_1_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_28_3"><i><font color="Green">E88</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_1_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_1_1_1_3"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_1_1_1_3"/><b>then </b>
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_1"><i><font color="Green">E105</font></i></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">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_1"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_1"><i><font color="Green">E103</font></i></a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>
<br/>.= 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan13.html#E41:1_1_1_28_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_1"><i><font color="Green">E101</font></i></a>, <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>
;<br/><b>hence </b><a NAME="E4:1_1_1_28_3_3_1_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1_28_3"><i><font color="Green">E89</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_1_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_1_1_1_4"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_1_1_1_4"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_1"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_1"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_1"><i><font color="Green">E105</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E31:1_1_1_28_3"><i><font color="Green">E92</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E41:1_1_1_28_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_1"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_1_1_1_4"><i><font color="Green">E108</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_28_3"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green">E86</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:1_1_1_28_3_3_1_1_1_4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_28_3"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E33:1_1_1_28_3"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jordan13.html#E36:1_1_1_28_3"><i><font color="Green">E94</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="1_1_1_28_3_3_2"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_2"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>26</sub></font>, <font color="Maroon">c<sub>27</sub></font>, <font color="Maroon">c<sub>28</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_1_28_3_3_2"/><i><font color="Green">E102</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:1_1_1_28_3_3_2"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E5:1_1_1_28_3_3_2"/><i><font color="Green">E104</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>28</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:1_1_1_28_3_3_2"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>28</sub></font>
 <b>and </b><br/><a NAME="E7:1_1_1_28_3_3_2"/><i><font color="Green">E106</font></i>: 
( ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E8:1_1_1_28_3_3_2"/><i><font color="Green">E107</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>26</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_2"><i><font color="Green">E102</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><b>thus </b><a NAME="E9:1_1_1_28_3_3_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_3_3_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_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="E1:1_1_1_28_3_3_2_1_1"/>
( ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3_3_2"><i><font color="Green">E106</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_2_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_2_1_1_1"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_2_1_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_2"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_2_1_1_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_2"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_2"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_2_1_1_1"><i><font color="Green">E108</font></i></a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>
<br/>.= 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan13.html#E41:1_1_1_28_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_1"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_2_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>
;<br/><b>hence </b><a NAME="E5:1_1_1_28_3_3_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1_28_3"><i><font color="Green">E89</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_2_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_2_1_1_2"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_2_1_1_2"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_2"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E4:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green">E110</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E5:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E6:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green">E112</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E57:1_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E7:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green">E113</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>21</sub></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">E114</font></i>: <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E111</font></i></a></i>
;<br/><a NAME="E9:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green">E115</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a></i>;<br/> <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan13.html#E41:1_1_1_28_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>
<br/>.= 
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E114</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E115</font></i></a>, <a class="ref" href="gobrd13.html#T38">GOBRD13:38</a></i>
;<br/><a NAME="E11:1_1_1_28_3_3_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R1">turns_right</a> <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E33:1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1_28_3"><i><font color="Green">E89</font></i></a></i>;<br/><a NAME="E12:1_1_1_28_3_3_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="gobrd13.html#R1">turns_right</a> <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E57:1_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_28_3"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E112</font></i></a>, <a class="ref" href="gobrd13.html#T44">GOBRD13:44</a></i>;<br/><a NAME="E13:1_1_1_28_3_3_2_1_1_2"/><b>then </b><i><font color="Green">E116</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>21</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>21</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_28_3"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E113</font></i></a>, <a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/><b>set </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>14</sub></font></span>)</span></span>)</span>;<br/><a NAME="E14:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green">E117</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jordan9.html#T29">JORDAN9:29</a></i>;<br/><a NAME="E15:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green">E118</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E16:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green">E119</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E76:1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E17:1_1_1_28_3_3_2_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green">E120</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> = 
<font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E16:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="finseq_5.html#T69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E30:1_1_1_28_3"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E119</font></i></a>, <a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>
;<br/><a NAME="E19:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_28_3_3_2"><i><font color="Green">E107</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E20:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green">E122</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E116</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E46:1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E65:1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E67:1_1_1"><i><font color="Green">E49</font></i></a>, <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">c<sub>14</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> <font color="Maroon">c<sub>14</sub></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">c<sub>14</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"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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="E22:1_1_1_28_3_3_2_1_1_2"/><b>then </b>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan13.html#E13:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E116</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>;<br/><a NAME="E23:1_1_1_28_3_3_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E43:1_1_1_28_3"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3_3_2"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E121</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E122</font></i></a>, <a class="ref" href="jordan9.html#T22">JORDAN9:22</a></i>;<br/><a NAME="E24:1_1_1_28_3_3_2_1_1_2"/><b>then </b><i><font color="Green">E123</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>14</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green">E90</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E25:1_1_1_28_3_3_2_1_1_2"/><i><font color="Green">E124</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">c<sub>14</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E87:1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="sppol_2.html#T22">SPPOL_2:22</a></i>;<br/><a NAME="E26:1_1_1_28_3_3_2_1_1_2"/>
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D1">GOBOARD9:def 1</a></i>;<br/><b>hence </b><a NAME="E27:1_1_1_28_3_3_2_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E89:1_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E123</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3_3_2_1_1_2"><i><font color="Green">E124</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_2_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_2_1_1_3"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_2_1_1_3"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_2"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_2"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E31:1_1_1_28_3"><i><font color="Green">E92</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E41:1_1_1_28_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_3"><i><font color="Green">E108</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_28_3"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green">E86</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>
;<br/><b>hence </b><a NAME="E4:1_1_1_28_3_3_2_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_28_3"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E33:1_1_1_28_3"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jordan13.html#E36:1_1_1_28_3"><i><font color="Green">E94</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_2_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_2_1_1_4"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_2_1_1_4"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_2"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_2"><i><font color="Green">E105</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan13.html#E41:1_1_1_28_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_2"><i><font color="Green">E101</font></i></a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>
<br/>.= 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_2"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_2"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_2_1_1_4"><i><font color="Green">E108</font></i></a>, <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>
;<br/><b>hence </b><a NAME="E4:1_1_1_28_3_3_2_1_1_4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_28_3"><i><font color="Green">E88</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="1_1_1_28_3_3_3"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_3"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>26</sub></font>, <font color="Maroon">c<sub>27</sub></font>, <font color="Maroon">c<sub>28</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_1_28_3_3_3"/><i><font color="Green">E102</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:1_1_1_28_3_3_3"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E5:1_1_1_28_3_3_3"/><i><font color="Green">E104</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>28</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:1_1_1_28_3_3_3"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>28</sub></font>
 <b>and </b><br/><a NAME="E7:1_1_1_28_3_3_3"/><i><font color="Green">E106</font></i>: 
( ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E8:1_1_1_28_3_3_3"/><i><font color="Green">E107</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>26</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_3"><i><font color="Green">E102</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><b>thus </b><a NAME="E9:1_1_1_28_3_3_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_3_3_3_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_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="E1:1_1_1_28_3_3_3_1_1"/>
( ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3_3_3"><i><font color="Green">E106</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_3_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_3_1_1_1"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_3_1_1_1"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_3"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_3"><i><font color="Green">E105</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_3_1_1_1"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green">E101</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E4:1_1_1_28_3_3_3_1_1_1"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>24</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_3_1_1_1"><i><font color="Green">E109</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>23</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan13.html#E41:1_1_1_28_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green">E101</font></i></a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>
<br/>.= 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_3"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_3"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_3"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_3"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3_1_1_1"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_3_1_1_1"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_3_1_1_1"><i><font color="Green">E111</font></i></a>, <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>
;<br/><b>hence </b><a NAME="E6:1_1_1_28_3_3_3_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_28_3"><i><font color="Green">E88</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_3_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_3_1_1_2"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_3_1_1_2"/><b>then </b>
( <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_3"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_3"><i><font color="Green">E105</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_3_1_1_2"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green">E101</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E31:1_1_1_28_3"><i><font color="Green">E92</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E41:1_1_1_28_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_3"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_3_1_1_2"><i><font color="Green">E108</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_28_3"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green">E86</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:1_1_1_28_3_3_3_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_28_3"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E33:1_1_1_28_3"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jordan13.html#E36:1_1_1_28_3"><i><font color="Green">E94</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_3_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_3_1_1_3"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_3_1_1_3"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_3"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_3"><i><font color="Green">E105</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green">E101</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E4:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E5:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E6:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green">E113</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E57:1_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E8:1_1_1_28_3_3_3_1_1_3"/>
<font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T31">XREAL_1:31</a></i>;<br/><a NAME="E9:1_1_1_28_3_3_3_1_1_3"/><b>then </b><i><font color="Green">E114</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>22</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E108</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><i><font color="Green">E115</font></i>: <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E112</font></i></a></i>
;<br/><a NAME="E11:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a></i>;<br/> <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan13.html#E41:1_1_1_28_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green">E101</font></i></a>, <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>
<br/>.= 
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_3"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_3"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E116</font></i></a>, <a class="ref" href="gobrd13.html#T40">GOBRD13:40</a></i>
;<br/><a NAME="E13:1_1_1_28_3_3_3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R1">turns_right</a> <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E33:1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1_28_3"><i><font color="Green">E89</font></i></a></i>;<br/><a NAME="E14:1_1_1_28_3_3_3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="gobrd13.html#R1">turns_right</a> <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E57:1_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_28_3"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E113</font></i></a>, <a class="ref" href="gobrd13.html#T44">GOBRD13:44</a></i>;<br/><a NAME="E15:1_1_1_28_3_3_3_1_1_3"/><b>then </b><i><font color="Green">E117</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>21</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></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">c<sub>21</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_28_3"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_3"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_3"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E114</font></i></a>, <a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/><a NAME="E16:1_1_1_28_3_3_3_1_1_3"/><b>then </b><i><font color="Green">E118</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>22</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>29</sub></font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>14</sub></font></span>)</span></span>)</span>;<br/><a NAME="E17:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green">E119</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jordan9.html#T29">JORDAN9:29</a></i>;<br/><a NAME="E18:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green">E120</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E76:1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E19:1_1_1_28_3_3_3_1_1_3"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green">E121</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> = 
<font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E18:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E120</font></i></a>, <a class="ref" href="finseq_5.html#T69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E30:1_1_1_28_3"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E120</font></i></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">c<sub>14</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E46:1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E65:1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E67:1_1_1"><i><font color="Green">E49</font></i></a>, <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">c<sub>14</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> <font color="Maroon">c<sub>14</sub></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">c<sub>14</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"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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="E22:1_1_1_28_3_3_3_1_1_3"/><b>then </b>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E121</font></i></a>, <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>;<br/><a NAME="E23:1_1_1_28_3_3_3_1_1_3"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E43:1_1_1_28_3"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_3"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3_3_3"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E118</font></i></a>, <a class="ref" href="jordan9.html#T22">JORDAN9:22</a></i>;<br/><a NAME="E24:1_1_1_28_3_3_3_1_1_3"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>14</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green">E90</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E25:1_1_1_28_3_3_3_1_1_3"/><i><font color="Green">E123</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">c<sub>14</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E87:1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="sppol_2.html#T22">SPPOL_2:22</a></i>;<br/><a NAME="E26:1_1_1_28_3_3_3_1_1_3"/>
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D1">GOBOARD9:def 1</a></i>;<br/><b>hence </b><a NAME="E27:1_1_1_28_3_3_3_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E89:1_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E119</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3_3_3_1_1_3"><i><font color="Green">E123</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_3_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_3_1_1_4"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_3_1_1_4"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_3"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_3"><i><font color="Green">E105</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_3_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green">E101</font></i></a>, <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">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_3"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_3"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_3_1_1_4"><i><font color="Green">E108</font></i></a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>
<br/>.= 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan13.html#E41:1_1_1_28_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_3"><i><font color="Green">E101</font></i></a>, <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>
;<br/><b>hence </b><a NAME="E5:1_1_1_28_3_3_3_1_1_4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1_28_3"><i><font color="Green">E89</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="1_1_1_28_3_3_4"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_4"/><i><font color="Green">E101</font></i>: 
( <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>26</sub></font>, <font color="Maroon">c<sub>27</sub></font>, <font color="Maroon">c<sub>28</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:1_1_1_28_3_3_4"/><i><font color="Green">E102</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E4:1_1_1_28_3_3_4"/><i><font color="Green">E103</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E5:1_1_1_28_3_3_4"/><i><font color="Green">E104</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>28</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E6:1_1_1_28_3_3_4"/><i><font color="Green">E105</font></i>: 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>28</sub></font>
 <b>and </b><br/><a NAME="E7:1_1_1_28_3_3_4"/><i><font color="Green">E106</font></i>: 
( ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E8:1_1_1_28_3_3_4"/><i><font color="Green">E107</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>25</sub></font> &amp; <font color="Maroon">c<sub>25</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>26</sub></font> &amp; <font color="Maroon">c<sub>26</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_4"><i><font color="Green">E102</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><b>thus </b><a NAME="E9:1_1_1_28_3_3_4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_28_3_3_4_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_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="E1:1_1_1_28_3_3_4_1_1"/>
( ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) or ( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3_3_4"><i><font color="Green">E106</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_4_1_1_1"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_4_1_1_1"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_4_1_1_1"/><b>then </b>
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_4"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_4"><i><font color="Green">E105</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_4_1_1_1"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green">E101</font></i></a></i>;<br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E31:1_1_1_28_3"><i><font color="Green">E92</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E41:1_1_1_28_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_4"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_4_1_1_1"><i><font color="Green">E108</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_28_3"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green">E86</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a></i>
;<br/><b>hence </b><a NAME="E5:1_1_1_28_3_3_4_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E42:1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan13.html#E17:1_1_1_28_3"><i><font color="Green">E82</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E33:1_1_1_28_3"><i><font color="Green">E93</font></i></a>, <a class="txt" href="jordan13.html#E36:1_1_1_28_3"><i><font color="Green">E94</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_4_1_1_2"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_4_1_1_2"/><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_4_1_1_2"/><b>then </b><i><font color="Green">E109</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_4"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_4"><i><font color="Green">E105</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_4_1_1_2"/><b>then </b><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green">E101</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E4:1_1_1_28_3_3_4_1_1_2"/><i><font color="Green">E111</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_4_1_1_2"><i><font color="Green">E109</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>25</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>26</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_4"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_4"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_4"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_4"><i><font color="Green">E105</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4_1_1_2"><i><font color="Green">E108</font></i></a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>
<br/>.= 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan13.html#E41:1_1_1_28_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_4_1_1_2"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_4_1_1_2"><i><font color="Green">E111</font></i></a>, <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>
;<br/><b>hence </b><a NAME="E6:1_1_1_28_3_3_4_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1_28_3"><i><font color="Green">E89</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_4_1_1_3"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_4_1_1_3"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_4_1_1_3"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_4"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_4"><i><font color="Green">E105</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_4_1_1_3"/><b>then </b><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green">E101</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/> <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>23</sub></font>,<font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan13.html#E41:1_1_1_28_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green">E101</font></i></a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>
<br/>.= 
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_4"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_4"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_4_1_1_3"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_4_1_1_3"><i><font color="Green">E109</font></i></a>, <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>
;<br/><b>hence </b><a NAME="E5:1_1_1_28_3_3_4_1_1_3"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E26:1_1_1_28_3"><i><font color="Green">E88</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="1_1_1_28_3_3_4_1_1_4"><b>suppose </b></a><a NAME="E1:1_1_1_28_3_3_4_1_1_4"/>
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:1_1_1_28_3_3_4_1_1_4"/><b>then </b><i><font color="Green">E108</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_4"><i><font color="Green">E104</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_4"><i><font color="Green">E105</font></i></a>, <a class="ref" href="goboard1.html#T21">GOBOARD1:21</a></i>;<br/><a NAME="E3:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green">E109</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green">E101</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/><a NAME="E4:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green">E110</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E5:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green">E111</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E15:1_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a></i>;<br/><a NAME="E6:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green">E112</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E7:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green">E113</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E57:1_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E8:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green">E114</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>22</sub></font> )
 <b>by </b><i><a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E108</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><i><font color="Green">E115</font></i>: <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E60:1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E112</font></i></a></i>
;<br/><a NAME="E10:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green">E116</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a></i>;<br/> <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>24</sub></font>
<b>by </b><i><a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3"><i><font color="Green">E83</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3"><i><font color="Green">E86</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3"><i><font color="Green">E87</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E40:1_1_1_28_3"><i><font color="Green">E97</font></i></a>, <a class="txt" href="jordan13.html#E41:1_1_1_28_3"><i><font color="Green">E98</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green">E101</font></i></a>, <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>
<br/>.= 
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_28_3_3_4"><i><font color="Green">E101</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_4"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_4"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E109</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E110</font></i></a>, <a class="txt" href="jordan13.html#E5:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E111</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E115</font></i></a>, <a class="txt" href="jordan13.html#E10:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E116</font></i></a>, <a class="ref" href="gobrd13.html#T42">GOBRD13:42</a></i>
;<br/><a NAME="E12:1_1_1_28_3_3_4_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="gobrd13.html#R1">turns_right</a> <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E33:1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E27:1_1_1_28_3"><i><font color="Green">E89</font></i></a></i>;<br/><a NAME="E13:1_1_1_28_3_3_4_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="gobrd13.html#R1">turns_right</a> <font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E57:1_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_28_3"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E113</font></i></a>, <a class="ref" href="gobrd13.html#T44">GOBRD13:44</a></i>;<br/><a NAME="E14:1_1_1_28_3_3_4_1_1_4"/><b>then </b><i><font color="Green">E117</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>22</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>22</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan13.html#E7:1_1_1_28_3"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E30:1_1_1_28_3"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E3:1_1_1_28_3_3_4"><i><font color="Green">E102</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_4"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E114</font></i></a>, <a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/><b>set </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>14</sub></font></span>)</span></span>)</span>;<br/><a NAME="E15:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green">E118</font></i>: 
<span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jordan9.html#T29">JORDAN9:29</a></i>;<br/><a NAME="E16:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green">E119</font></i>: 
2 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E76:1_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E17:1_1_1_28_3_3_4_1_1_4"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</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">c<sub>14</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1_28_3"><i><font color="Green">E85</font></i></a>, <a class="ref" href="finseq_5.html#D3">FINSEQ_5:def 3</a></i>;<br/><b>then </b><i><font color="Green">E120</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> = 
<font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E16:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E119</font></i></a>, <a class="ref" href="finseq_5.html#T69">FINSEQ_5:69</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="txt" href="jordan13.html#E30:1_1_1_28_3"><i><font color="Green">E91</font></i></a>, <a class="txt" href="jordan13.html#E16:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E119</font></i></a>, <a class="ref" href="finseq_5.html#T30">FINSEQ_5:30</a></i>
;<br/><i><font color="Green">E121</font></i>: <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 = 
<font color="Maroon">c<sub>14</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E46:1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan13.html#E65:1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jordan13.html#E67:1_1_1"><i><font color="Green">E49</font></i></a>, <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">c<sub>14</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> <font color="Maroon">c<sub>14</sub></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">c<sub>14</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"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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="E20:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green">E122</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E117</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E21:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green">E123</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E43:1_1_1_28_3"><i><font color="Green">E100</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E22:1_1_1_28_3_3_4_1_1_4"/><b>then </b>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>21</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E1:1_1_1_28_3"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3"><i><font color="Green">E72</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_28_3"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E38:1_1_1_28_3"><i><font color="Green">E95</font></i></a>, <a class="txt" href="jordan13.html#E39:1_1_1_28_3"><i><font color="Green">E96</font></i></a>, <a class="txt" href="jordan13.html#E14:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E117</font></i></a>, <a class="txt" href="jordan13.html#E18:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E120</font></i></a>, <a class="txt" href="jordan13.html#E19:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E121</font></i></a>, <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>;<br/><a NAME="E23:1_1_1_28_3_3_4_1_1_4"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E43:1_1_1_28_3"><i><font color="Green">E100</font></i></a>, <a class="txt" href="jordan13.html#E4:1_1_1_28_3_3_4"><i><font color="Green">E103</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3_3_4"><i><font color="Green">E107</font></i></a>, <a class="txt" href="jordan13.html#E2:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E108</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E122</font></i></a>, <a class="txt" href="jordan13.html#E21:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E123</font></i></a>, <a class="ref" href="jordan9.html#T22">JORDAN9:22</a></i>;<br/><a NAME="E24:1_1_1_28_3_3_4_1_1_4"/><b>then </b><i><font color="Green">E124</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <span class="p2">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>14</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green">E90</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E25:1_1_1_28_3_3_4_1_1_4"/><i><font color="Green">E125</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">c<sub>14</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E87:1_1_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="sppol_2.html#T22">SPPOL_2:22</a></i>;<br/><a NAME="E26:1_1_1_28_3_3_4_1_1_4"/>
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></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">c<sub>14</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D1">GOBOARD9:def 1</a></i>;<br/><b>hence </b><a NAME="E27:1_1_1_28_3_3_4_1_1_4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="goboard9.html#K1">Rev</a> <font color="Maroon">c<sub>14</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E51:1_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan13.html#E89:1_1_1"><i><font color="Green">E65</font></i></a>, <a class="txt" href="jordan13.html#E8:1_1_1_28_3"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E9:1_1_1_28_3"><i><font color="Green">E76</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_28_3"><i><font color="Green">E90</font></i></a>, <a class="txt" href="jordan13.html#E15:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E118</font></i></a>, <a class="txt" href="jordan13.html#E24:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E124</font></i></a>, <a class="txt" href="jordan13.html#E25:1_1_1_28_3_3_4_1_1_4"><i><font color="Green">E125</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="E6:1_1_1_28"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="ref" href="goboard9.html#T26">GOBOARD9:26</a></i>;<br/>
<a NAME="E7:1_1_1_28"/><b>then </b>
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E82:1_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E88:1_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan13.html#E90:1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="jordan9.html#T3">JORDAN9:3</a></i>;<br/>
<b>hence </b><a NAME="E8:1_1_1_28"/>
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">E67</font></i>: <font color="Maroon">c<sub>14</sub></font> = 
<font color="Maroon">c<sub>11</sub></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">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="finseq_5.html#T31">FINSEQ_5:31</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>11</sub></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> ;<br/>
<a NAME="E94:1_1_1"/>
<font color="Maroon">c<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>then </b><i><font color="Green">E68</font></i>: <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> 1</span>)</span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan13.html#E41:1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a></i>
;<br/>
<a NAME="E96:1_1_1"/>
<font color="Maroon">c<sub>5</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E6:1_1_1"><i><font color="Green">E6</font></i></a></i>;<br/>
<b>then </b><i><font color="Green">E69</font></i>: <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> 2</span>)</span> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>
<br/>.= 
<font color="Maroon">c<sub>16</sub></font> <a href="finseq_4.html#K4">/.</a> 2
<b>by </b><i><a class="txt" href="jordan13.html#E55:1_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan13.html#E80:1_1_1"><i><font color="Green">E59</font></i></a></i>
;<br/>
<a NAME="E98:1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> is <a href="sprect_2.html#V1">clockwise_oriented</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1_32"><b>proof </b></a><div class="add">

<a NAME="E1:1_1_1_32"/><i><font color="Green">E70</font></i>: 
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>16</sub></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">c<sub>16</sub></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></i>;<br/>
<a NAME="E2:1_1_1_32"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>16</sub></font> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T73">JORDAN2C:73</a></i>;<br/>
<a NAME="E3:1_1_1_32"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></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 <br/>( <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R2">is_outside_component_of</a>  <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan2c.html#K2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan2c.html#T76">JORDAN2C:76</a></i>;<br/>
<a NAME="E4:1_1_1_32"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="jordan2c.html#K2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>16</sub></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> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#D4">JORDAN2C:def 4</a></i>;<br/>
<a NAME="E5:1_1_1_32"/><i><font color="Green">E72</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">c<sub>16</sub></font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">c<sub>16</sub></font>,1,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E6:1_1_1_32"/><i><font color="Green">E73</font></i>: 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1"><i><font color="Green">E1</font></i></a>, <a class="ref" href="jordan11.html#T6">JORDAN11:6</a></i>;<br/>
<a NAME="E7:1_1_1_32"/><i><font color="Green">E74</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E17:1_1_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/>
<a NAME="E8:1_1_1_32"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E1:1"><i><font color="Green">E1</font></i></a>, <a class="ref" href="jordan11.html#T9">JORDAN11:9</a></i>;<br/>
<a NAME="E9:1_1_1_32"/><b>then </b>
 <a href="gobrd13.html#K3">right_cell</a> <font color="Maroon">c<sub>16</sub></font>,1,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E19:1_1_1"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E54:1_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan13.html#E95:1_1_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="jordan13.html#E97:1_1_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="jordan13.html#E6:1_1_1_32"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E7:1_1_1_32"><i><font color="Green">E74</font></i></a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/>
<a NAME="E10:1_1_1_32"/><b>then </b><i><font color="Green">E75</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">c<sub>16</sub></font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E5:1_1_1_32"><i><font color="Green">E72</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E11:1_1_1_32"/><i><font color="Green">E76</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">c<sub>16</sub></font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E54:1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="jordan1h.html#T31">JORDAN1H:31</a></i>;<br/>
<a NAME="E12:1_1_1_32"/>
 <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">c<sub>16</sub></font>,1,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E54:1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="jordan9.html#T11">JORDAN9:11</a></i>;<br/>
<a NAME="E13:1_1_1_32"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>1</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E10:1_1_1_32"><i><font color="Green">E75</font></i></a>, <a class="txt" href="jordan13.html#E11:1_1_1_32"><i><font color="Green">E76</font></i></a>, <a class="ref" href="xboole_1.html#T68">XBOOLE_1:68</a></i>;<br/>
<a NAME="E14:1_1_1_32"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E82:1_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan13.html#E90:1_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan13.html#E92:1_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan13.html#E1:1_1_1_32"><i><font color="Green">E70</font></i></a>, <a class="ref" href="goboard9.html#T6">GOBOARD9:6</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <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> : <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> ;<br/>
<a NAME="E15:1_1_1_32"/>
 <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <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> : <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan2c.html#D5">JORDAN2C:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:1_1_1_32"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <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> : <font color="Olive">b<sub>1</sub></font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">c<sub>1</sub></font> </span>}</span> 
 <b>and </b><br/><a NAME="E18:1_1_1_32"/><i><font color="Green">E80</font></i>: 
 <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E13:1_1_1_32"><i><font color="Green">E77</font></i></a>, <a class="ref" href="zfmisc_1.html#T98">ZFMISC_1:98</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>19</sub></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><b> such that </b><br/><a NAME="E20:1_1_1_32"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E21:1_1_1_32"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="jordan2c.html#R1">is_inside_component_of</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E17:1_1_1_32"><i><font color="Green">E79</font></i></a></i>;<br/>
<a NAME="E22:1_1_1_32"/><i><font color="Green">E83</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="txt" href="jordan13.html#E21:1_1_1_32"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jordan2c.html#D3">JORDAN2C:def 3</a></i>;<br/>
<a NAME="E23:1_1_1_32"/>
 <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="ref" href="gobrd14.html#T24">GOBRD14:24</a></i>;<br/>
<a NAME="E24:1_1_1_32"/><b>then </b>
 <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>16</sub></font> <a href="subset_1.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E14:1_1_1_32"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E25:1_1_1_32"/><b>then </b>
 <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font> <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="E26:1_1_1_32"/><b>then </b><i><font color="Green">E84</font></i>: 
 <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>16</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>19</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E18:1_1_1_32"><i><font color="Green">E80</font></i></a>, <a class="txt" href="jordan13.html#E20:1_1_1_32"><i><font color="Green">E81</font></i></a>, <a class="txt" href="jordan13.html#E22:1_1_1_32"><i><font color="Green">E83</font></i></a>, <a class="ref" href="goboard9.html#T6">GOBOARD9:6</a></i>;<br/>
<a NAME="E27:1_1_1_32"/>
<font color="Maroon">c<sub>19</sub></font> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i><a class="txt" href="jordan13.html#E21:1_1_1_32"><i><font color="Green">E82</font></i></a>, <a class="ref" href="jordan2c.html#D3">JORDAN2C:def 3</a></i>;<br/>
<a NAME="E28:1_1_1_32"/><b>then </b><i><font color="Green">E85</font></i>: 
 <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>16</sub></font> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i><a class="txt" href="jordan13.html#E26:1_1_1_32"><i><font color="Green">E84</font></i></a>, <a class="ref" href="jordan2c.html#T16">JORDAN2C:16</a></i>;<br/>
<a NAME="E29:1_1_1_32"/><i><font color="Green">E86</font></i>: 
(  <a href="jordan2c.html#K2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard9.html#K3">RightComp</a> <font color="Maroon">c<sub>16</sub></font> or  <a href="jordan2c.html#K2">UBD</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard9.html#K2">LeftComp</a> <font color="Maroon">c<sub>16</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E4:1_1_1_32"><i><font color="Green">E71</font></i></a>, <a class="ref" href="jordan1h.html#T30">JORDAN1H:30</a></i>;<br/>
<a NAME="E30:1_1_1_32"/>
 <a href="topreal1.html#K5">L~</a> <font color="Maroon">c<sub>16</sub></font> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T73">JORDAN2C:73</a></i>;<br/>
<b>hence </b><a NAME="E31:1_1_1_32"/>
<font color="Maroon">c<sub>16</sub></font> is <a href="sprect_2.html#V1">clockwise_oriented</a>
 <b>by </b><i><a class="txt" href="jordan13.html#E28:1_1_1_32"><i><font color="Green">E85</font></i></a>, <a class="txt" href="jordan13.html#E29:1_1_1_32"><i><font color="Green">E86</font></i></a>, <a class="ref" href="jordan1h.html#T47">JORDAN1H:47</a>, <a class="ref" href="jordan1h.html#T49">JORDAN1H:49</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>16</sub></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">c<sub>17</sub></font>
;<br/>

<b>thus </b><a NAME="E100:1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E23:1_1_1"><i><font color="Green">E19</font></i></a></i>;<br/>

<b>thus </b><a NAME="E101:1_1_1"/>
<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E95:1_1_1"><i><font color="Green">E68</font></i></a></i>;<br/>

<b>thus </b><a NAME="E102:1_1_1"/>
<font color="Maroon">c<sub>17</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan13.html#E97:1_1_1"><i><font color="Green">E69</font></i></a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>18</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E103:1_1_1"/><i><font color="Green">E70</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></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">c<sub>17</sub></font> )
 ;<br/>

<a NAME="E104:1_1_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>18</sub></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="E105:1_1_1"/><b>then </b><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>18</sub></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">c<sub>17</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E103:1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E106:1_1_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E107:1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T39">BINARITH:39</a></i>;<br/>
<a NAME="E108:1_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>18</sub></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="txt" href="jordan13.html#E103:1_1_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E109:1_1_1"/><i><font color="Green">E75</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>18</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>18</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan13.html#E50:1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan13.html#E103:1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E105:1_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="jordan13.html#E106:1_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="gobrd13.html#T43">GOBRD13:43</a></i>;<br/>
<a NAME="E110:1_1_1"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>17</sub></font> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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="txt" href="jordan13.html#E32:1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan13.html#E47:1_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan13.html#E103:1_1_1"><i><font color="Green">E70</font></i></a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:1_1_1_33"/>
(  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><a NAME="E2:1_1_1_33"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>18</sub></font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E33:1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan13.html#E107:1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E108:1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E109:1_1_1"><i><font color="Green">E75</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_33"/>
<font color="Maroon">c<sub>17</sub></font> <a href="gobrd13.html#R2">turns_left</a> <font color="Maroon">c<sub>18</sub></font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E103:1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E110:1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="gobrd13.html#T45">GOBRD13:45</a></i>;<br/></div>
<b>end;</b></div>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:1_1_1_34"/>
(  <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><a NAME="E2:1_1_1_34"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>18</sub></font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E33:1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan13.html#E107:1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E108:1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E109:1_1_1"><i><font color="Green">E75</font></i></a></i>;<br/><b>hence </b><a NAME="E3:1_1_1_34"/>
<font color="Maroon">c<sub>17</sub></font> <a href="gobrd13.html#R3">goes_straight</a> <font color="Maroon">c<sub>18</sub></font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E103:1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E110:1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="gobrd13.html#T46">GOBRD13:46</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E113:1_1_1"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>18</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E114:1_1_1"/><b>then </b>
<font color="Maroon">c<sub>5</sub></font> <a href="gobrd13.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>18</sub></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">c<sub>18</sub></font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan13.html#E33:1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan13.html#E107:1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jordan13.html#E108:1_1_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="jordan13.html#E109:1_1_1"><i><font color="Green">E75</font></i></a></i>;<br/>
<b>hence </b><a NAME="E115:1_1_1"/>
<font color="Maroon">c<sub>17</sub></font> <a href="gobrd13.html#R1">turns_right</a> <font color="Maroon">c<sub>18</sub></font>, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan13.html#E103:1_1_1"><i><font color="Green">E70</font></i></a>, <a class="txt" href="jordan13.html#E110:1_1_1"><i><font color="Green">E76</font></i></a>, <a class="ref" href="gobrd13.html#T44">GOBRD13:44</a></i>;<br/>


</div>
