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

<b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>set </b><font color="Maroon">G</font> =  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>;<br/>
<b>assume </b><a NAME="E1:9"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">n</font> <a href="jordan1h.html#R1">is_sufficiently_large_for</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:9"/><b>then </b><i><font color="Green">E38</font></i>: 
(  <a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan13.html#D1">JORDAN13:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> holds <br/>(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Olive">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Olive">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> );<br/>
<a NAME="E3:9"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<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">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E44</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> 0</span>)</span> )
 ;<br/>

<a NAME="E2:9_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E3:9_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> 0</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:9_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> 0
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>hence </b><a NAME="E5:9_1"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> 0</span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> 0</span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:9"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

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

<a NAME="E1:9_2"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="goboard1.html#T38">GOBOARD1:38</a></i>;<br/>
<a NAME="E2:9_2"/><i><font color="Green">E54</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="goboard1.html#T38">GOBOARD1:38</a></i>;<br/>
<b>assume </b><a NAME="E3:9_2"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> &amp; <font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> holds <br/>(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<font color="Olive">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<font color="Olive">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/>

<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">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> or <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">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"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:9_2_1_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E3:9_2_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><b>hence </b><a NAME="E4:9_2_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><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">E57</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:9_2_1_2"/><b>then </b><i><font color="Green">E58</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:9_2_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E4:9_2_1_2"/><i><font color="Green">E67</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E5:9_2_1_2"/><i><font color="Green">E68</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="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">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E69</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E3:9_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>hence </b><a NAME="E4:9_2_1_2_1_1_1"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i/>;<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">E70</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon">i</font> =  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>;<br/><b>set </b><font color="Maroon">j</font> =  <a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>;<br/><a NAME="E2:9_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i>, , , <a class="ref" href="jordan8.html#T1">JORDAN8:1</a></i>;<br/><a NAME="E3:9_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E71</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/><a NAME="E4:9_2_1_2_1_1_2"/><i><font color="Green">E72</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="jordan11.html#T8">JORDAN11:8</a>, <a class="ref" href="jordan11.html#T9">JORDAN11:9</a></i>;<br/><a NAME="E5:9_2_1_2_1_1_2"/>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E6:9_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , , <a class="ref" href="xxreal_0.html#T1">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">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E74</font></i>: 
(  <a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:9_2_1_2_1_1_2"/><b>then </b>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#D2">GOBRD13:def 2</a></i>;<br/><b>hence </b><a NAME="E10:9_2_1_2_1_1_2"/>
 <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="jordan11.html#T11">JORDAN11:11</a></i>;<br/><a NAME="E11:9_2_1_2_1_1_2"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i>, , , , , , <a class="ref" href="gobrd13.html#D3">GOBRD13:def 3</a></i>;<br/><b>hence </b><a NAME="E12:9_2_1_2_1_1_2"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="jordan11.html#T10">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">E75</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3"/><b>then </b><i><font color="Green">E76</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E3:9_2_1_2_1_1_3"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E4:9_2_1_2_1_1_3"/><b>then </b><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E5:9_2_1_2_1_1_3"/><i><font color="Green">E79</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="finseq_5.html#T18">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">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> or <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E80</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><b>consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1"/><i><font color="Green">E81</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> )
 <b>and </b><br/><a NAME="E4:9_2_1_2_1_1_3_1_1_1"/><i><font color="Green">E82</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E5:9_2_1_2_1_1_3_1_1_1"/>
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , , , , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E6:9_2_1_2_1_1_3_1_1_1"/><i><font color="Green">E83</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E7:9_2_1_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E84</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font>
 <b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_3_1_1_1"/><i><font color="Green">E85</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E9:9_2_1_2_1_1_3_1_1_1"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E10:9_2_1_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E86</font></i>: 
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> ) or (  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> ) or  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E87</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> <a href="gobrd13.html#R2">turns_left</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , , , <a class="ref" href="jordan13.html#D1">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">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> ) )
 </span><b>by </b><i>, , , , <a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E89</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E90</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <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">E91</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="gobrd13.html#T35">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">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T27">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">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> )
 <b>by </b><i>, , , , , , , , , <a class="ref" href="binarith.html#T39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T22">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">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T26">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">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">E92</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E93</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <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">E94</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="gobrd13.html#T37">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">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T23">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">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> )
 <b>by </b><i>, , , , , , , , , <a class="ref" href="binarith.html#T39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T24">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">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T22">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">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">E95</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/><i><font color="Green">E96</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <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">E97</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 ;<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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="gobrd13.html#T39">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">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T29">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">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , , <a class="ref" href="gobrd13.html#T26">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">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T28">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">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">E98</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/><i><font color="Green">E99</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <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">E100</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="gobrd13.html#T41">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">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T25">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">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , , <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></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">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T24">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">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E101</font></i>: 
(  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E102</font></i>: 
 <a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> <a href="gobrd13.html#R3">goes_straight</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , , , <a class="ref" href="jordan13.html#D1">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">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) )
 </span><b>by </b><i>, , , , <a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E103</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E104</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <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">E105</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="gobrd13.html#T36">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">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T23">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">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , , <a class="ref" href="gobrd13.html#T35">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">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T22">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">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">E106</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E107</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <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">E108</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="gobrd13.html#T38">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">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T25">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">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j1</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , , <a class="ref" href="gobrd13.html#T37">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">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T24">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">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">E109</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/><i><font color="Green">E110</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <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">E111</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="gobrd13.html#T40">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">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T27">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">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , , <a class="ref" href="gobrd13.html#T39">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">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T26">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">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">E112</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/><i><font color="Green">E113</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <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">E114</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 ;<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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>by </b><i>, , , , , , , , , <a class="ref" href="binarith.html#T53">BINARITH:53</a>, <a class="ref" href="gobrd13.html#T42">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">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T29">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">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , , <a class="ref" href="gobrd13.html#T41">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">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T28">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">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E115</font></i>: 
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E116</font></i>: 
 <a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font> <a href="gobrd13.html#R1">turns_right</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 <b>by </b><i>, , , , , <a class="ref" href="jordan13.html#D1">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">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> ) )
 </span><b>by </b><i>, , , , <a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E117</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><i><font color="Green">E118</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <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">E119</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j1</font> )
 <b>by </b><i>, , , , , , , , <a class="ref" href="binarith.html#T39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T23">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">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T25">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">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , , <a class="ref" href="gobrd13.html#T36">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">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T24">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">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">E120</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/><i><font color="Green">E121</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <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">E122</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 ;<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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i1</font> )
 <b>by </b><i>, , , , , , , , <a class="ref" href="binarith.html#T39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T25">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">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T29">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">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , , <a class="ref" href="gobrd13.html#T38">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">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T28">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">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">E123</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/><i><font color="Green">E124</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<span class="p2">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <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">E125</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="gobrd13.html#T27">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">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T23">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">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , , <a class="ref" href="gobrd13.html#T40">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">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T22">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">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">E126</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/><i><font color="Green">E127</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <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">E128</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font>
 ;<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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="gobrd13.html#T29">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">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T27">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">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , , <a class="ref" href="gobrd13.html#T42">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">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , , , , <a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , <a class="ref" href="gobrd13.html#T26">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">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T32">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">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E12:9_2_1_2_1_1_3_1_1_1"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:9_2_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E129</font></i>: 
<font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:9_2_1_2_1_1_3_1_1_2"/><i><font color="Green">E130</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> )
 <b>and </b><br/><a NAME="E6:9_2_1_2_1_1_3_1_1_2"/><i><font color="Green">E131</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 <b>and </b><br/><a NAME="E7:9_2_1_2_1_1_3_1_1_2"/><i><font color="Green">E132</font></i>: 
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i>, , <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_3_1_1_2"/><i><font color="Green">E133</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E9:9_2_1_2_1_1_3_1_1_2"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:9_2_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E134</font></i>: 
( <font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> &amp; <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> )
 <b>by </b><i>, , , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:9_2_1_2_1_1_3_1_1_2"/><i><font color="Green">E135</font></i>: 
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , </i>;<br/><a NAME="E12:9_2_1_2_1_1_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i>, <a class="ref" href="jordan8.html#T2">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">E136</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">m</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font> )
 <b>by </b><i>, , , <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="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">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E137</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_2_1_1_1"/><b>then </b>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font> )
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E3:9_2_1_2_1_1_3_1_1_2_1_1_1"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , , , , , <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E138</font></i>: 
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_2_1_1_2"/><b>then </b>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font> )
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a>, <a class="ref" href="gobrd13.html#T25">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">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , , , , , <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E139</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_2_1_1_3"/><b>then </b>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E3:9_2_1_2_1_1_3_1_1_2_1_1_3"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , , , , , <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E140</font></i>: 
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_2_1_1_4"/><b>then </b>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j2</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">k</font></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j2</font> )
 <b>by </b><i>, , , , , <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/><b>hence </b><a NAME="E3:9_2_1_2_1_1_3_1_1_2_1_1_4"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , , , , , <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E15:9_2_1_2_1_1_3_1_1_2"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:9_2_1_2_1_1_3"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E7:9_2_1_2"/>
(  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">m</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:9"/><i><font color="Green">E141</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<a NAME="E6:9"/>
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/>
<b>hence </b><a NAME="E7:9"/>
(  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or  not <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> or (  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">C</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font> ) )
 <b>by </b><i/>;<br/>


</div>
