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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>



<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>assume </b><a NAME="E1:9"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="jordan1h.html#R1">is_sufficiently_large_for</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:9"/><b>then </b><i><font color="Green">E8</font></i>: 
(  <a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan13.html#D1">JORDAN13:def 1</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> );<br/>
<a NAME="E3:9"/><i><font color="Green">E9</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">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:9_1"/><i><font color="Green">E10</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> 0</span>)</span>
 
<b>by </b><i><a class="txt" href="jordan14.html#E1:9_1"><i><font color="Green">E10</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> 0</span>)</span>,<font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> 0</span>)</span>,<font color="Maroon">c<sub>6</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:9"/><i><font color="Green">E10</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_2"><b>proof </b></a><div class="add">

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

<a NAME="E1:9_2"/><i><font color="Green">E11</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="ref" href="goboard1.html#T38">GOBOARD1:38</a></i>;<br/>
<a NAME="E2:9_2"/><i><font color="Green">E12</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="ref" href="goboard1.html#T38">GOBOARD1:38</a></i>;<br/>
<b>assume </b><a NAME="E3:9_2"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Olive">b<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<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">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> or <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:9_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9_2_1_1"><i><font color="Green">E14</font></i></a>, <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">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan14.html#E3:9_2"><i><font color="Green">E13</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2"><b>suppose </b></a><a NAME="E1:9_2_1_2"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:9_2_1_2"/><b>then </b><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:9_2_1_2"/><b>then </b><i><font color="Green">E16</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></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">E17</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E5:9_2_1_2"/><i><font color="Green">E18</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="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">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 )
 </span><b>by </b><i><a class="ref" href="cqc_the1.html#T2">CQC_THE1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E19</font></i>: 
<font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E3:9_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:9_2_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E5:9_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:9_2_1_2_1_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><a NAME="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_4.html#K2">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_2"><i><font color="Green">E19</font></i></a>, <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">E20</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/><a NAME="E4:9_2_1_2_1_1_2"/><i><font color="Green">E21</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9"><i><font color="Green">E7</font></i></a>, <a class="ref" href="jordan11.html#T8">JORDAN11:8</a>, <a class="ref" href="jordan11.html#T9">JORDAN11:9</a></i>;<br/><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">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <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>
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E7:9_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E8:9_2_1_2_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="binarith.html#T52">BINARITH:52</a></i>;<br/><a NAME="E9:9_2_1_2_1_1_2"/><b>then </b><i><font color="Green">E23</font></i>: 
(  <a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1 )
 <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_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E7:9_2_1_2_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="gobrd13.html#D2">GOBRD13:def 2</a></i>;<br/><b>hence </b><a NAME="E11:9_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9"><i><font color="Green">E7</font></i></a>, <a class="ref" href="jordan11.html#T11">JORDAN11:11</a></i>;<br/><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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan1h.html#K3">X-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan11.html#K3">Y-SpanStart</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_2"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_2"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E7:9_2_1_2_1_1_2"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan14.html#E9:9_2_1_2_1_1_2"><i><font color="Green">E23</font></i></a>, <a class="ref" href="gobrd13.html#D3">GOBRD13:def 3</a></i>;<br/><b>hence </b><a NAME="E13:9_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9"><i><font color="Green">E7</font></i></a>, <a class="ref" href="jordan11.html#T10">JORDAN11:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E19</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3"/><b>then </b><i><font color="Green">E20</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <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">E21</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <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">E22</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></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">E23</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> or <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E24</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></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_1"/><b>then </b><i><font color="Green">E25</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:9_2_1_2_1_1_3_1_1_1"/><i><font color="Green">E26</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>and </b><br/><a NAME="E5:9_2_1_2_1_1_3_1_1_1"/><i><font color="Green">E27</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E6:9_2_1_2_1_1_3_1_1_1"/>
( ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E7:9_2_1_2_1_1_3_1_1_1"/><i><font color="Green">E28</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E8:9_2_1_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E9:9_2_1_2_1_1_3_1_1_1"/><i><font color="Green">E30</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E7:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E10:9_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E3:9_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a></i>;<br/><a NAME="E11:9_2_1_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E31</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green">E23</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> ) 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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> ) or  <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 </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">E32</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3"><i><font color="Green">E22</font></i></a>, <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">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="gobrd13.html#D7">GOBRD13:def 7</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E34</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <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">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E8:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E11:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>;<br/><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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E8:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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">E34</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>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">E36</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E11:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>;<br/><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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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">E34</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>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">E36</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E9:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E11:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E9:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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">E34</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <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">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E11:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E32</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3"><i><font color="Green">E22</font></i></a>, <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">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="ref" href="gobrd13.html#D8">GOBRD13:def 8</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E34</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <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">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>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">E36</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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">E34</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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">E34</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E8:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E8:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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">E34</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <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">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>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">E36</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E7:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E9:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E32</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:9_2_1_2_1_1_3_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1, <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3"><i><font color="Green">E22</font></i></a>, <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">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3"><i><font color="Green">E33</font></i></a>, <a class="ref" href="gobrd13.html#D6">GOBRD13:def 6</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E34</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 <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">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E11:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/><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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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">E34</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>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">E36</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E11:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="binarith.html#T39">BINARITH:39</a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/><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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E9:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E9:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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">E34</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>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">E36</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E11:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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">E34</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <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">E35</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 2</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3"><i><font color="Green">E23</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E11:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E8:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green">E34</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan14.html#E8:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green">E36</font></i></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_1"><i><font color="Green">E24</font></i></a>, <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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:9_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <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">E24</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E3:9_2_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan14.html#E4:9_2_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:9_2_1_2_1_1_3_1_1_2"/><i><font color="Green">E25</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>and </b><br/><a NAME="E6:9_2_1_2_1_1_3_1_1_2"/><i><font color="Green">E26</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 <b>and </b><br/><a NAME="E7:9_2_1_2_1_1_3_1_1_2"/><i><font color="Green">E27</font></i>: 
( ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <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">E28</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E9:9_2_1_2_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E24</font></i></a>, <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">E29</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan14.html#E8:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E28</font></i></a>, <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">E30</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E3:9_2"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E24</font></i></a></i>;<br/><a NAME="E12:9_2_1_2_1_1_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="finseq_1.html#K12">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9_2_1_2"><i><font color="Green">E15</font></i></a>, <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">E31</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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E10:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="finseq_4.html#T83">FINSEQ_4:83</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="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">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> ) or ( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan14.html#E7:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E27</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="9_2_1_2_1_1_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:9_2_1_2_1_1_3_1_1_2_1_1_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> )
 ;<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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E3:9_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E11:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan14.html#E13:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_2_1_1_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E32</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> &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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/><b>hence </b><a NAME="E3:9_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E11:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan14.html#E13:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_2_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E32</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E3:9_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E11:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan14.html#E13:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_2_1_1_3"><i><font color="Green">E32</font></i></a>, <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="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">E32</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>11</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>11</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E2:9_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E3:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/><b>hence </b><a NAME="E3:9_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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan14.html#E1:9_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan14.html#E5:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan14.html#E6:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan14.html#E11:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan14.html#E13:9_2_1_2_1_1_3_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan14.html#E1:9_2_1_2_1_1_3_1_1_2_1_1_4"><i><font color="Green">E32</font></i></a>, <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>7</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E5:9"/><i><font color="Green">E11</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="jordan14.html#E3:9"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan14.html#E4:9"><i><font color="Green">E10</font></i></a>);<br/></i>
<a NAME="E6:9"/>
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="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">c<sub>3</sub></font> or  not <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="jordan14.html#E5:9"><i><font color="Green">E11</font></i></a></i>;<br/>


</div>
