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

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon" title="c2">n</font>, <font color="Maroon" title="c3">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>set </b><font color="Maroon" title="c4">G</font> =  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>;<br/>
<b>set </b><font color="Maroon" title="c5">f</font> =  <a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>;<br/>
<b>assume </b><a NAME="E1:9"/><i><font color="Green" title="E6">A1</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="jordan1h.html#R1" title="JORDAN1H:pred.1">is_sufficiently_large_for</a> <font color="Maroon" title="c1">C</font>
 ;<br/>

<a NAME="E2:9"/><b>then </b><i><font color="Green" title="E7">A2</font></i>: 
(  <a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan13.html#D1" title="JORDAN13:def.1">JORDAN13:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b>for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1</span>)</span> holds <br/>(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1</span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> $1</span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> );<br/>
<a NAME="E3:9"/><i><font color="Green" title="E8">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green" title="E9">A4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> &amp; <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> )
 ;<br/>

<a NAME="E2:9_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<a NAME="E3:9_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="jordan14.html#E1:9_1"><i><font color="Green" title="E9">A4</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:9_1"/><b>then </b>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="card_1.html#T47" title="CARD_1:th.47">CARD_1:47</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c6">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>,<font color="Maroon" title="c6">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:9"/><i><font color="Green" title="E9">A5</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">k</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:9_2"/><i><font color="Green" title="E10">A6</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="goboard1.html#T38" title="GOBOARD1:th.38">GOBOARD1:38</a></i>;<br/>
<a NAME="E2:9_2"/><i><font color="Green" title="E11">A7</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a>  <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="goboard1.html#T38" title="GOBOARD1:th.38">GOBOARD1:38</a></i>;<br/>
<b>assume </b><a NAME="E3:9_2"/><i><font color="Green" title="E12">A8</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> &amp; <font color="Olive" title="b1">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span> holds <br/>(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Olive" title="b1">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:9_2_1"/>
( <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_1"><b>suppose </b></a><a NAME="E1:9_2_1_1"/>
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:9_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> )
 <b>by </b><i><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a>, <a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><b>hence </b><a NAME="E3:9_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan14.html#E3:9_2"><i><font color="Green" title="E12">A8</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2"><b>suppose </b></a><a NAME="E1:9_2_1_2"/><i><font color="Green" title="E13">A9</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:9_2_1_2"/><b>then </b><i><font color="Green" title="E14">A10</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:9_2_1_2"/><b>then </b><i><font color="Green" title="E15">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><a NAME="E4:9_2_1_2"/><i><font color="Green" title="E16">A12</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">k</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="ref" href="finseq_1.html#T80" title="FINSEQ_1:th.80">FINSEQ_1:80</a></i>;<br/><b>let </b><font color="Maroon" title="c7">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E5:9_2_1_2"/><i><font color="Green" title="E17">A13</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font> &amp; <font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_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="E1:9_2_1_2_1_1"/>
( <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:9_2_1_2_1_1_1"/><i><font color="Green" title="E18">A14</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E3:9_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_1"><i><font color="Green" title="E18">A14</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E4:9_2_1_2_1_1_1"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:9_2_1_2_1_1_2"/><i><font color="Green" title="E18">A15</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c8">i</font> =  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>;<br/><b>set </b><font color="Maroon" title="c9">j</font> =  <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>;<br/><a NAME="E2:9_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_2"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="jordan8.html#T1" title="JORDAN8:th.1">JORDAN8:1</a></i>;<br/><a NAME="E3:9_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E19">A16</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> 2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T26" title="FINSEQ_4:th.26">FINSEQ_4:26</a></i>;<br/><a NAME="E4:9_2_1_2_1_1_2"/><i><font color="Green" title="E20">A17</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="jordan11.html#T8" title="JORDAN11:th.8">JORDAN11:8</a>, <a class="ref" href="jordan11.html#T9" title="JORDAN11:th.9">JORDAN11:9</a></i>;<br/><a NAME="E5:9_2_1_2_1_1_2"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/><a NAME="E6:9_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E21">A18</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_2"><i><font color="Green" title="E18">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E7:9_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="ref" href="binarith.html#T52" title="BINARITH:th.52">BINARITH:52</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_2"/><b>then </b><i><font color="Green" title="E22">A19</font></i>: 
(  <a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E9:9_2_1_2_1_1_2"/><b>then </b>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_2"><i><font color="Green" title="E19">A16</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_2"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_2"><i><font color="Green" title="E21">A18</font></i></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></i>;<br/><b>hence </b><a NAME="E10:9_2_1_2_1_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="jordan11.html#T11" title="JORDAN11:th.11">JORDAN11:11</a></i>;<br/><a NAME="E11:9_2_1_2_1_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3" title="JORDAN1H:func.3">X-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K3" title="JORDAN11:func.3">Y-SpanStart</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_2"><i><font color="Green" title="E19">A16</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_2"><i><font color="Green" title="E20">A17</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_2"><i><font color="Green" title="E21">A18</font></i></a>, <a class="txt" href="jordan14.html#E8:9_2_1_2_1_1_2"><i><font color="Green" title="E22">A19</font></i></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></i>;<br/><b>hence </b><a NAME="E12:9_2_1_2_1_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="jordan11.html#T10" title="JORDAN11:th.10">JORDAN11:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:9_2_1_2_1_1_3"/><i><font color="Green" title="E18">A20</font></i>: 
<font color="Maroon" title="c6">k</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E19">A21</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="ref" href="binarith.html#T67" title="BINARITH:th.67">BINARITH:67</a></i>;<br/><a NAME="E3:9_2_1_2_1_1_3"/><i><font color="Green" title="E20">A22</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E4:9_2_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E21">A23</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><a NAME="E5:9_2_1_2_1_1_3"/><i><font color="Green" title="E22">A24</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T18" title="FINSEQ_5:th.18">FINSEQ_5:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1_2_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_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="E1:9_2_1_2_1_1_3_1_1"/>
( <font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> or <font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:9_2_1_2_1_1_3_1_1_1"/><i><font color="Green" title="E23">A25</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c8">i1</font>, <font color="Maroon" title="c9">j1</font>, <font color="Maroon" title="c10">i2</font>, <font color="Maroon" title="c11">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1"/><i><font color="Green" title="E24">A26</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font> )
 <b>and </b><br/><a NAME="E4:9_2_1_2_1_1_3_1_1_1"/><i><font color="Green" title="E25">A27</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font> )
 <b>and </b><br/><a NAME="E5:9_2_1_2_1_1_3_1_1_1"/>
( ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E6:9_2_1_2_1_1_3_1_1_1"/><i><font color="Green" title="E26">A28</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c10">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">j2</font> &amp; <font color="Maroon" title="c11">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/><a NAME="E7:9_2_1_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E27">A29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font>
 <b>by </b><i><a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_3_1_1_1"/><i><font color="Green" title="E28">A30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E26">A28</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/><a NAME="E9:9_2_1_2_1_1_3_1_1_1"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E3:9_2"><i><font color="Green" title="E12">A8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a></i>;<br/><a NAME="E10:9_2_1_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green" title="E29">A31</font></i>: 
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green" title="E22">A24</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_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="E1:9_2_1_2_1_1_3_1_1_1_1_1"/>
( (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) or  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:9_2_1_2_1_1_3_1_1_1_1_1_1"/><i><font color="Green" title="E30">A32</font></i>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E31">A33</font></i>: 
 <a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> <a href="gobrd13.html#R2" title="GOBRD13:pred.2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3"><i><font color="Green" title="E21">A23</font></i></a>, <a class="ref" href="jordan13.html#D1" title="JORDAN13:def.1">JORDAN13:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_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="E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E31">A33</font></i></a>, <a class="ref" href="gobrd13.html#D7" title="GOBRD13:def.7">GOBRD13:def 7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E32">A34</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E33">A35</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><i><font color="Green" title="E34">A36</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>
 ;<br/><div class="add"><a NAME="E4:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E30">A32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E32">A34</font></i></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></i>;<br/><a NAME="E5:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E7:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E32">A34</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E33">A35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E34">A36</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E6:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/><a NAME="E7:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/>
(  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c9">j1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E10:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E29">A31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E32">A34</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E7:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E32">A34</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E33">A35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green" title="E34">A36</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>;<br/><b>hence </b><a NAME="E9:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E32">A37</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E33">A38</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/><i><font color="Green" title="E34">A39</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E4:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E30">A32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E32">A37</font></i></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></i>;<br/><a NAME="E5:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/><b>then </b>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E33">A38</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E34">A39</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E6:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/><a NAME="E7:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/>
(  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c11">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E10:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E29">A31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E32">A37</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E32">A37</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E33">A38</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green" title="E34">A39</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>;<br/><b>hence </b><a NAME="E9:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/><i><font color="Green" title="E32">A40</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/><i><font color="Green" title="E33">A41</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/><i><font color="Green" title="E34">A42</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E4:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E30">A32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E32">A40</font></i></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></i>;<br/><a NAME="E5:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/><b>then </b>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E8:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E28">A30</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E33">A41</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E34">A42</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/><b>hence </b><a NAME="E6:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/><a NAME="E7:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E10:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E29">A31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E32">A40</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E8:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E28">A30</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E33">A41</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green" title="E34">A42</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>;<br/><b>hence </b><a NAME="E9:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/><i><font color="Green" title="E32">A43</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/><i><font color="Green" title="E33">A44</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/><i><font color="Green" title="E34">A45</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>
 ;<br/><div class="add"><a NAME="E4:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1"><i><font color="Green" title="E30">A32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green" title="E32">A43</font></i></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></i>;<br/><a NAME="E5:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/><b>then </b>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green" title="E32">A43</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green" title="E33">A44</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green" title="E34">A45</font></i></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/><b>hence </b><a NAME="E6:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/><a NAME="E7:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c11">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E10:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E29">A31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green" title="E32">A43</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green" title="E32">A43</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green" title="E33">A44</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green" title="E34">A45</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>;<br/><b>hence </b><a NAME="E9:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:9_2_1_2_1_1_3_1_1_1_1_1_1"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:9_2_1_2_1_1_3_1_1_1_1_1_2"/><i><font color="Green" title="E30">A46</font></i>: 
(  <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K6" title="GOBRD13:func.6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_2"/><b>then </b><i><font color="Green" title="E31">A47</font></i>: 
 <a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> <a href="gobrd13.html#R3" title="GOBRD13:pred.3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3"><i><font color="Green" title="E21">A23</font></i></a>, <a class="ref" href="jordan13.html#D1" title="JORDAN13:def.1">JORDAN13:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_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="E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1"/>
( ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green" title="E31">A47</font></i></a>, <a class="ref" href="gobrd13.html#D8" title="GOBRD13:def.8">GOBRD13:def 8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E32">A48</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E33">A49</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/><i><font color="Green" title="E34">A50</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E4:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green" title="E30">A46</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E32">A48</font></i></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></i>;<br/><a NAME="E5:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/><b>then </b>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E32">A48</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E33">A49</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E34">A50</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E6:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/><a NAME="E7:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green" title="E30">A46</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E32">A48</font></i></a>, <a class="ref" href="gobrd13.html#T35" title="GOBRD13:th.35">GOBRD13:35</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E33">A49</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green" title="E34">A50</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>;<br/><b>hence </b><a NAME="E9:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E32">A51</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E33">A52</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/><i><font color="Green" title="E34">A53</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>
 ;<br/><div class="add"><a NAME="E4:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green" title="E30">A46</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E32">A51</font></i></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></i>;<br/><a NAME="E5:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/><b>then </b>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E32">A51</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E33">A52</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E34">A53</font></i></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/><b>hence </b><a NAME="E6:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/><a NAME="E7:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c9">j1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green" title="E30">A46</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E32">A51</font></i></a>, <a class="ref" href="gobrd13.html#T37" title="GOBRD13:th.37">GOBRD13:37</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E32">A51</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E33">A52</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green" title="E34">A53</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>;<br/><b>hence </b><a NAME="E9:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/><i><font color="Green" title="E32">A54</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/><i><font color="Green" title="E33">A55</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/><i><font color="Green" title="E34">A56</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>
 ;<br/><div class="add"><a NAME="E4:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green" title="E30">A46</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green" title="E32">A54</font></i></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></i>;<br/><a NAME="E5:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/><b>then </b>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E7:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green" title="E33">A55</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green" title="E34">A56</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E6:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/><a NAME="E7:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green" title="E30">A46</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green" title="E32">A54</font></i></a>, <a class="ref" href="gobrd13.html#T39" title="GOBRD13:th.39">GOBRD13:39</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E7:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green" title="E33">A55</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green" title="E34">A56</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>;<br/><b>hence </b><a NAME="E9:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/><i><font color="Green" title="E32">A57</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/><i><font color="Green" title="E33">A58</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/><i><font color="Green" title="E34">A59</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E4:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/>
(  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E26">A28</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green" title="E30">A46</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E32">A57</font></i></a>, <a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></i>;<br/><a NAME="E5:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/><b>then </b>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E33">A58</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E34">A59</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/><b>hence </b><a NAME="E6:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/><a NAME="E7:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green" title="E30">A46</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E32">A57</font></i></a>, <a class="ref" href="gobrd13.html#T41" title="GOBRD13:th.41">GOBRD13:41</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E8:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E28">A30</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E33">A58</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green" title="E34">A59</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>;<br/><b>hence </b><a NAME="E9:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:9_2_1_2_1_1_3_1_1_1_1_1_2"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:9_2_1_2_1_1_3_1_1_1_1_1_3"/><i><font color="Green" title="E30">A60</font></i>: 
 <a href="gobrd13.html#K5" title="GOBRD13:func.5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E31">A61</font></i>: 
 <a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font> <a href="gobrd13.html#R1" title="GOBRD13:pred.1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1, <a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9"><i><font color="Green" title="E6">A1</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3"><i><font color="Green" title="E21">A23</font></i></a>, <a class="ref" href="jordan13.html#D1" title="JORDAN13:def.1">JORDAN13:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_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="E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1"/>
( ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font> ) )
 </span><b>by </b><i><a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3"><i><font color="Green" title="E31">A61</font></i></a>, <a class="ref" href="gobrd13.html#D6" title="GOBRD13:def.6">GOBRD13:def 6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><i><font color="Green" title="E32">A62</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><i><font color="Green" title="E33">A63</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><i><font color="Green" title="E34">A64</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>
 ;<br/><div class="add"><a NAME="E4:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/>
(  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">j1</font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E10:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E29">A31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E32">A62</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/><a NAME="E5:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><b>then </b>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E32">A62</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E33">A63</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E34">A64</font></i></a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/><b>hence </b><a NAME="E6:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/><a NAME="E7:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3"><i><font color="Green" title="E30">A60</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E32">A62</font></i></a>, <a class="ref" href="gobrd13.html#T36" title="GOBRD13:th.36">GOBRD13:36</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E33">A63</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green" title="E34">A64</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a></i>;<br/><b>hence </b><a NAME="E9:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/><i><font color="Green" title="E32">A65</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/><i><font color="Green" title="E33">A66</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/><i><font color="Green" title="E34">A67</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E4:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/>
(  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">i1</font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E10:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E29">A31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E32">A65</font></i></a>, <a class="ref" href="binarith.html#T39" title="BINARITH:th.39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/><a NAME="E5:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/><b>then </b>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E8:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E28">A30</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E32">A65</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E33">A66</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E34">A67</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/><b>hence </b><a NAME="E6:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/><a NAME="E7:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3"><i><font color="Green" title="E30">A60</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E32">A65</font></i></a>, <a class="ref" href="gobrd13.html#T38" title="GOBRD13:th.38">GOBRD13:38</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E8:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E28">A30</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E33">A66</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green" title="E34">A67</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a></i>;<br/><b>hence </b><a NAME="E9:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/><i><font color="Green" title="E32">A68</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/><i><font color="Green" title="E33">A69</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/><i><font color="Green" title="E34">A70</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E4:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E10:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E29">A31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green" title="E32">A68</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/><a NAME="E5:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/><b>then </b>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green" title="E33">A69</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green" title="E34">A70</font></i></a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E6:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/><a NAME="E7:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3"><i><font color="Green" title="E30">A60</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green" title="E32">A68</font></i></a>, <a class="ref" href="gobrd13.html#T40" title="GOBRD13:th.40">GOBRD13:40</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green" title="E33">A69</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green" title="E34">A70</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a></i>;<br/><b>hence </b><a NAME="E9:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/><i><font color="Green" title="E32">A71</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/><i><font color="Green" title="E33">A72</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/><i><font color="Green" title="E34">A73</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font>
 ;<br/><div class="add"><a NAME="E4:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green" title="E22">A24</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E10:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E29">A31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green" title="E32">A71</font></i></a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/><a NAME="E5:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/><b>then </b>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E7:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green" title="E33">A72</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green" title="E34">A73</font></i></a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E6:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/>
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/><a NAME="E7:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green" title="E13">A9</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green" title="E19">A21</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E24">A26</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3"><i><font color="Green" title="E30">A60</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green" title="E32">A71</font></i></a>, <a class="ref" href="gobrd13.html#T42" title="GOBRD13:th.42">GOBRD13:42</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/><b>then </b>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green" title="E18">A20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green" title="E20">A22</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E25">A27</font></i></a>, <a class="txt" href="jordan14.html#E7:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E27">A29</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green" title="E33">A72</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green" title="E34">A73</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a></i>;<br/><b>hence </b><a NAME="E9:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green" title="E7">A2</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green" title="E23">A25</font></i></a>, <a class="ref" href="gobrd13.html#T32" title="GOBRD13:th.32">GOBRD13:32</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:9_2_1_2_1_1_3_1_1_1_1_1_3"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:9_2_1_2_1_1_3_1_1_1"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:9_2_1_2_1_1_3_1_1_2"/>
<font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:9_2_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E23">A74</font></i>: 
<font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green" title="E15">A11</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green" title="E16">A12</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>then consider </b><font color="Maroon" title="c8">i1</font>, <font color="Maroon" title="c9">j1</font>, <font color="Maroon" title="c10">i2</font>, <font color="Maroon" title="c11">j2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E5:9_2_1_2_1_1_3_1_1_2"/><i><font color="Green" title="E24">A75</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font> )
 <b>and </b><br/><a NAME="E6:9_2_1_2_1_1_3_1_1_2"/><i><font color="Green" title="E25">A76</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font> )
 <b>and </b><br/><a NAME="E7:9_2_1_2_1_1_3_1_1_2"/><i><font color="Green" title="E26">A77</font></i>: 
( ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9_2"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_3_1_1_2"/><i><font color="Green" title="E27">A78</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E9:9_2_1_2_1_1_3_1_1_2"/>
<font color="Maroon" title="c7">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E23">A74</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E10:9_2_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E28">A79</font></i>: 
( <font color="Maroon" title="c7">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span> &amp; <font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E23">A74</font></i></a>, <a class="txt" href="jordan14.html#E8:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E27">A78</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:9_2_1_2_1_1_3_1_1_2"/><i><font color="Green" title="E29">A80</font></i>: 
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E3:9_2"><i><font color="Green" title="E12">A8</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E23">A74</font></i></a></i>;<br/><a NAME="E12:9_2_1_2_1_1_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12" title="FINSEQ_1:func.12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green" title="E14">A10</font></i></a>, <a class="ref" href="jordan8.html#T2" title="JORDAN8:th.2">JORDAN8:2</a></i>;<br/><a NAME="E13:9_2_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green" title="E30">A81</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">m</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E24">A75</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E25">A76</font></i></a>, <a class="txt" href="jordan14.html#E10:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E28">A79</font></i></a>, <a class="ref" href="finseq_4.html#T83" title="FINSEQ_4:th.83">FINSEQ_4:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_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="E1:9_2_1_2_1_1_3_1_1_2_1_1"/>
( ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> ) or ( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan14.html#E7:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E26">A77</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:9_2_1_2_1_1_3_1_1_2_1_1_1"/><i><font color="Green" title="E31">A82</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_2_1_1_1"/><b>then </b>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c8">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c9">j1</font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9_2"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E23">A74</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E24">A75</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E25">A76</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E3:9_2_1_2_1_1_3_1_1_2_1_1_1"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E24">A75</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E25">A76</font></i></a>, <a class="txt" href="jordan14.html#E11:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E29">A80</font></i></a>, <a class="txt" href="jordan14.html#E13:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E30">A81</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_2_1_1_1"><i><font color="Green" title="E31">A82</font></i></a>, <a class="ref" href="gobrd13.html#T22" title="GOBRD13:th.22">GOBRD13:22</a>, <a class="ref" href="gobrd13.html#T23" title="GOBRD13:th.23">GOBRD13:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:9_2_1_2_1_1_3_1_1_2_1_1_2"/><i><font color="Green" title="E31">A83</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_2_1_1_2"/><b>then </b>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c9">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c9">j1</font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9_2"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E23">A74</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E24">A75</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E25">A76</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/><b>hence </b><a NAME="E3:9_2_1_2_1_1_3_1_1_2_1_1_2"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E24">A75</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E25">A76</font></i></a>, <a class="txt" href="jordan14.html#E11:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E29">A80</font></i></a>, <a class="txt" href="jordan14.html#E13:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E30">A81</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_2_1_1_2"><i><font color="Green" title="E31">A83</font></i></a>, <a class="ref" href="gobrd13.html#T24" title="GOBRD13:th.24">GOBRD13:24</a>, <a class="ref" href="gobrd13.html#T25" title="GOBRD13:th.25">GOBRD13:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:9_2_1_2_1_1_3_1_1_2_1_1_3"/><i><font color="Green" title="E31">A84</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> )
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_2_1_1_3"/><b>then </b>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<font color="Maroon" title="c11">j2</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c10">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c11">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9_2"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E23">A74</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E24">A75</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E25">A76</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E3:9_2_1_2_1_1_3_1_1_2_1_1_3"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E24">A75</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E25">A76</font></i></a>, <a class="txt" href="jordan14.html#E11:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E29">A80</font></i></a>, <a class="txt" href="jordan14.html#E13:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E30">A81</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_2_1_1_3"><i><font color="Green" title="E31">A84</font></i></a>, <a class="ref" href="gobrd13.html#T26" title="GOBRD13:th.26">GOBRD13:26</a>, <a class="ref" href="gobrd13.html#T27" title="GOBRD13:th.27">GOBRD13:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:9_2_1_2_1_1_3_1_1_2_1_1_4"/><i><font color="Green" title="E31">A85</font></i>: 
( <font color="Maroon" title="c8">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">i2</font> &amp; <font color="Maroon" title="c9">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_2_1_1_4"/><b>then </b>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c11">j2</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <font color="Maroon" title="c6">k</font></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c8">i1</font>,<font color="Maroon" title="c11">j2</font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9_2"><i><font color="Green" title="E11">A7</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E23">A74</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E24">A75</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E25">A76</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/><b>hence </b><a NAME="E3:9_2_1_2_1_1_3_1_1_2_1_1_4"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9_2"><i><font color="Green" title="E10">A6</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green" title="E17">A13</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E24">A75</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E25">A76</font></i></a>, <a class="txt" href="jordan14.html#E11:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E29">A80</font></i></a>, <a class="txt" href="jordan14.html#E13:9_2_1_2_1_1_3_1_1_2"><i><font color="Green" title="E30">A81</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_2_1_1_4"><i><font color="Green" title="E31">A85</font></i></a>, <a class="ref" href="gobrd13.html#T28" title="GOBRD13:th.28">GOBRD13:28</a>, <a class="ref" href="gobrd13.html#T29" title="GOBRD13:th.29">GOBRD13:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E15:9_2_1_2_1_1_3_1_1_2"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:9_2_1_2_1_1_3"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:9_2_1_2"/>
(  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c6">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c7">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:9"/><i><font color="Green" title="E10">A86</font></i>: 
for <font color="Olive" title="b1">k</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="jordan14.html#E3:9"><i><font color="Green" title="E8">A3</font></i></a>, <a class="txt" href="jordan14.html#E4:9"><i><font color="Green" title="E9">A5</font></i></a>);<br/></i>
<a NAME="E6:9"/>
<span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="finseq_1.html#K17" title="FINSEQ_1:func.17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T79" title="FINSEQ_1:th.79">FINSEQ_1:79</a></i>;<br/>
<b>hence </b><a NAME="E7:9"/>
(  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">k</font> or  not <font color="Maroon" title="c3">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> or (  <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c3">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c1">C</font> ) )
 <b>by </b><i><a class="txt" href="jordan14.html#E5:9"><i><font color="Green" title="E10">A86</font></i></a></i>;<br/>


</div>
