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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> non <a href="sppol_1.html#V1">horizontal</a> non <a href="sppol_1.html#V2">vertical</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>





<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<b>assume </b><a NAME="E1:34"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</sub></font> is <a href="connsp_1.html#V2">connected</a>
 ;<br/>

<a NAME="E2:34"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="jordan9.html#D1" target="_self">Def1</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>3</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>3</sub></font> );<br/>
<a NAME="E3:34"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E4:34"/><i><font color="Green">E35</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 3 &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 
<b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/>
<a NAME="E5:34"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> 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="34_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:34_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> 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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> holds <br/>(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:34_1_1"/>
( <font color="Maroon">c<sub>10</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> or <font color="Maroon">c<sub>10</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="34_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>10</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:34_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="E3:34_1_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:34_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#T79">FINSEQ_1:79</a></i>;<br/><b>hence </b><a NAME="E4:34_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="jordan9.html#E1:34_1"><i><font color="Green">E37</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>10</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 ;<br/><div class="add"><a NAME="E2:34_1_1_2"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>10</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="E3:34_1_1_2"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="ref" href="goboard1.html#T38">GOBOARD1:38</a></i>;<br/><a NAME="E4:34_1_1_2"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="ref" href="goboard1.html#T38">GOBOARD1:38</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:34_1_1_2"/><i><font color="Green">E42</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>and </b><br/><a NAME="E7:34_1_1_2"/><i><font color="Green">E43</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E8:34_1_1_2"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>and </b><br/><a NAME="E9:34_1_1_2"/><i><font color="Green">E45</font></i>: 
(  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> &amp;  <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:34"><i><font color="Green">E32</font></i></a>, <a class="ref" href="jordan9.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E10:34_1_1_2"/><i><font color="Green">E46</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:34_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E11:34_1_1_2"/><i><font color="Green">E47</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="finseq_1.html#T80">FINSEQ_1:80</a></i>;<br/><a NAME="E12:34_1_1_2"/><i><font color="Green">E48</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:34"><i><font color="Green">E35</font></i></a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E13:34_1_1_2"/>
( 2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1 )
 <b>by </b><i><a class="ref" href="heine.html#T7">HEINE:7</a>, <a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E14:34_1_1_2"/><b>then </b><i><font color="Green">E49</font></i>: 
2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E15:34_1_1_2"/><i><font color="Green">E50</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</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="34_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_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="E1:34_1_1_2_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1 or <font color="Maroon">c<sub>10</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="34_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_1_2_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:34_1_1_2_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E3:34_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E4:34_1_1_2_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><b>hence </b><a NAME="E5:34_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_1_2_1_1_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:34_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p3">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span></span><a href="finseq_4.html#K2">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E7:34_1_1_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="jordan9.html#E8:34_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="jordan8.html#T1">JORDAN8:1</a></i>;<br/><a NAME="E3:34_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E52</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="matrix_1.html#K1">width</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="ref" href="finseq_4.html#T26">FINSEQ_4:26</a></i>;<br/><a NAME="E4:34_1_1_2_1_1_2"/><i><font color="Green">E53</font></i>: 
( <font color="Maroon">c<sub>11</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="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E6:34_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:34_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E4:34"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan9.html#E6:34_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan9.html#E12:34_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><a NAME="E6:34_1_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>12</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:34_1_1_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> 1 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_2"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E8:34_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> 1
 ;<br/><a NAME="E9:34_1_1_2_1_1_2"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:34_1_1_2_1_1_2"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:34"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan9.html#E4:34_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E5:34_1_1_2_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan9.html#E8:34_1_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="gobrd13.html#D3">GOBRD13:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:34_1_1_2_1_1_2_1"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:34_1_1_2_1_1_2_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E10:34_1_1_2_1_1_2"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_2_1"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E5:34_1_1_2_1_1_2_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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>11</sub></font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="euclid.html#K23">]|</a></span> where B is   <a href="real_1.html#NM1">Real</a>, B is   <a href="real_1.html#NM1">Real</a> : ( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan9.html#E4:34"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan9.html#E6:34_1_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan9.html#E4:34_1_1_2_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="gobrd11.html#T31">GOBRD11:31</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E7:34_1_1_2_1_1_2_1"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="euclid.html#K23">]|</a></span>
 <b>and </b><br/><a NAME="E8:34_1_1_2_1_1_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>11</sub></font>,1</span>)</span> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21">`1</a>  )
 <b>and </b><br/><a NAME="E9:34_1_1_2_1_1_2_1"/><i><font color="Green">E60</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_2_1"><i><font color="Green">E58</font></i></a></i>;<br/><a NAME="E10:34_1_1_2_1_1_2_1"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default">1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:34"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan9.html#E12:34_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="goboard7.html#T10">GOBOARD7:10</a></i>;<br/><a NAME="E11:34_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default">1 <a href="real_1.html#K5">-</a> 2</span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p0">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p5">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p0">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 2</span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/><a NAME="E12:34_1_1_2_1_1_2_1"/>
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 2 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E4:34"><i><font color="Green">E35</font></i></a></i>;<br/><b>then </b><i><font color="Green">E62</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 2</span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan9.html#E14:34_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="xcmplx_1.html#T88">XCMPLX_1:88</a></i>
;<br/><a NAME="E14:34_1_1_2_1_1_2_1"/><i><font color="Green">E63</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> 1,<span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p4">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="real_1.html#K5">-</a> 2</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E11:34_1_1_2_1_1_2_1"><i><font color="Green">E61</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/><a NAME="E15:34_1_1_2_1_1_2_1"/>
 <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="jordan8.html#T12">JORDAN8:12</a></i>;<br/><a NAME="E16:34_1_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E17:34_1_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan9.html#E14:34_1_1_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><a NAME="E18:34_1_1_2_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default">2 <a href="card_4.html#K1">|^</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E19:34_1_1_2_1_1_2_1"/><b>then </b><i><font color="Green">E64</font></i>: 
 <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E9:34_1_1_2_1_1_2_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan9.html#E13:34_1_1_2_1_1_2_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan9.html#E14:34_1_1_2_1_1_2_1"><i><font color="Green">E63</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E20:34_1_1_2_1_1_2_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_2_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/><b>hence </b><a NAME="E21:34_1_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E7:34_1_1_2_1_1_2_1"><i><font color="Green">E59</font></i></a>, <a class="txt" href="jordan9.html#E19:34_1_1_2_1_1_2_1"><i><font color="Green">E64</font></i></a>, <a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:34_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 ;<br/><a NAME="E13:34_1_1_2_1_1_2"/><i><font color="Green">E58</font></i>: 
 <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="sprect_1.html#T13">SPRECT_1:13</a></i>;<br/><a NAME="E14:34_1_1_2_1_1_2"/>
 <a href="pscomp_1.html#K29">N-min</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E4:34"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan9.html#E4:34_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E9:34_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E5:34_1_1_2_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan9.html#E8:34_1_1_2_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E9:34_1_1_2_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="gobrd13.html#D2">GOBRD13:def 2</a></i>;<br/><b>hence </b><a NAME="E15:34_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E13:34_1_1_2_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:34_1_1_2_1_1_3"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1
 ;<br/><div class="add"><a NAME="E2:34_1_1_2_1_1_3"/><b>then </b><i><font color="Green">E52</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="binarith.html#K5">-'</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan3.html#T12">JORDAN3:12</a></i>;<br/><a NAME="E3:34_1_1_2_1_1_3"/><i><font color="Green">E53</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3"><i><font color="Green">E51</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_1_2_1_1_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_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="E1:34_1_1_2_1_1_3_1_1"/>
( <font color="Maroon">c<sub>12</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> or <font color="Maroon">c<sub>12</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</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="34_1_1_2_1_1_3_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_1_2_1_1_3_1_1_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>12</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:34_1_1_2_1_1_3_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>12</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E3:34_1_1_2_1_1_3_1_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="34_1_1_2_1_1_3_1_1_1_1"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E2:34_1_1_2_1_1_3_1_1_1_1"/><i><font color="Green">E57</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> )
 <b>and </b><br/><a NAME="E3:34_1_1_2_1_1_3_1_1_1_1"/><i><font color="Green">E58</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> )
 <b>and </b><br/><a NAME="E4:34_1_1_2_1_1_3_1_1_1_1"/>
( ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E5:34_1_1_2_1_1_3_1_1_1_1"/><i><font color="Green">E59</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <font color="Maroon">c<sub>16</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/><a NAME="E6:34_1_1_2_1_1_3_1_1_1_1"/><i><font color="Green">E60</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E7:34_1_1_2_1_1_3_1_1_1_1"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</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>15</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E5:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E8:34_1_1_2_1_1_3_1_1_1_1"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E5:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E9:34_1_1_2_1_1_3_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:34_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a></i>;<br/><a NAME="E10:34_1_1_2_1_1_3_1_1_1_1"/><b>then </b><i><font color="Green">E63</font></i>: 
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="gobrd13.html#T32">GOBRD13:32</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_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="E1:34_1_1_2_1_1_3_1_1_1_1_1"/>
( (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font> ) or (  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font> ) or  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2_1_1_3_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_1_2_1_1_3_1_1_1_1_1_1"/><i><font color="Green">E64</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font> )
 ;<br/><div class="add"><a NAME="E2:34_1_1_2_1_1_3_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:34"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E6:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jordan9.html#D1" target="_self">Def1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_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="34_1_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="E1:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1"/>
( ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E6:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_1"><i><font color="Green">E65</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="34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>and </b><br/><a NAME="E2:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E67</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>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><a NAME="E4:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T36">GOBRD13:36</a></i>;<br/><a NAME="E5:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>;<br/><b>hence </b><a NAME="E6:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T32">GOBRD13:32</a></i>;<br/><a NAME="E7:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp;  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E66</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="E8:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/><b>hence </b><a NAME="E9:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</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="34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>and </b><br/><a NAME="E2:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E4:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T38">GOBRD13:38</a></i>;<br/><a NAME="E5:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E8:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>;<br/><b>hence </b><a NAME="E6:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T32">GOBRD13:32</a></i>;<br/><a NAME="E7:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</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>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E66</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="E8:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E8:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/><b>hence </b><a NAME="E9:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</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="34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>and </b><br/><a NAME="E2:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E4:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T40">GOBRD13:40</a></i>;<br/><a NAME="E5:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>;<br/><b>hence </b><a NAME="E6:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T32">GOBRD13:32</a></i>;<br/><a NAME="E7:34_1_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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/><a NAME="E8:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E9:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</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="34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/><i><font color="Green">E67</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>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><a NAME="E4:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T42">GOBRD13:42</a></i>;<br/><a NAME="E5:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E7:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>;<br/><b>hence </b><a NAME="E6:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T32">GOBRD13:32</a></i>;<br/><a NAME="E7:34_1_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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/><a NAME="E8:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E7:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_1_1_1_4"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E9:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</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="E4:34_1_1_2_1_1_3_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2_1_1_3_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_1_2_1_1_3_1_1_1_1_1_2"/><i><font color="Green">E64</font></i>: 
(  <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font> )
 ;<br/><div class="add"><a NAME="E2:34_1_1_2_1_1_3_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:34"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E6:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jordan9.html#D1" target="_self">Def1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_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="34_1_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="E1:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1"/>
( ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</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>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</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>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E6:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E65</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="34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>and </b><br/><a NAME="E2:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E4:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T35">GOBRD13:35</a></i>;<br/><a NAME="E5:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>;<br/><b>hence </b><a NAME="E6:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T32">GOBRD13:32</a></i>;<br/><a NAME="E7:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T36">GOBRD13:36</a></i>;<br/><a NAME="E8:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E9:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</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="34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>and </b><br/><a NAME="E2:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E67</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>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><a NAME="E4:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T37">GOBRD13:37</a></i>;<br/><a NAME="E5:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>;<br/><b>hence </b><a NAME="E6:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T32">GOBRD13:32</a></i>;<br/><a NAME="E7:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T38">GOBRD13:38</a></i>;<br/><a NAME="E8:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/><b>hence </b><a NAME="E9:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</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="34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>and </b><br/><a NAME="E2:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/><i><font color="Green">E67</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>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><a NAME="E4:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T39">GOBRD13:39</a></i>;<br/><a NAME="E5:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E7:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>;<br/><b>hence </b><a NAME="E6:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T32">GOBRD13:32</a></i>;<br/><a NAME="E7:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T40">GOBRD13:40</a></i>;<br/><a NAME="E8:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E7:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E9:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</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="34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E4:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T41">GOBRD13:41</a></i>;<br/><a NAME="E5:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E8:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>;<br/><b>hence </b><a NAME="E6:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T32">GOBRD13:32</a></i>;<br/><a NAME="E7:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"/>
 <a href="gobrd13.html#K5">front_right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T42">GOBRD13:42</a></i>;<br/><a NAME="E8:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E8:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_2"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_2_1_1_4"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/><b>hence </b><a NAME="E9:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</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="E4:34_1_1_2_1_1_3_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2_1_1_3_1_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:34_1_1_2_1_1_3_1_1_1_1_1_3"/><i><font color="Green">E64</font></i>: 
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font>
 ;<br/><div class="add"><a NAME="E2:34_1_1_2_1_1_3_1_1_1_1_1_3"/><b>then </b><i><font color="Green">E65</font></i>: 
 <a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:34"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E6:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="ref" href="jordan9.html#D1" target="_self">Def1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1_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="34_1_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="E1:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1"/>
( ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E6:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_3"><i><font color="Green">E65</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="34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>and </b><br/><a NAME="E2:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><i><font color="Green">E67</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>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><a NAME="E4:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/>
( <font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> &amp;  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green">E66</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="E5:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E7:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>;<br/><b>hence </b><a NAME="E6:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T32">GOBRD13:32</a></i>;<br/><a NAME="E7:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T35">GOBRD13:35</a></i>;<br/><a NAME="E8:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E7:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E9:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</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="34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>and </b><br/><a NAME="E2:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E4:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/>
( <font color="Maroon">c<sub>15</sub></font> <a href="binarith.html#K5">-'</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> &amp;  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green">E66</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="E5:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a></i>;<br/><b>hence </b><a NAME="E6:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T32">GOBRD13:32</a></i>;<br/><a NAME="E7:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T37">GOBRD13:37</a></i>;<br/><a NAME="E8:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_2"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E9:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</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="34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>and </b><br/><a NAME="E2:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/><i><font color="Green">E67</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>16</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 ;<br/><div class="add"><a NAME="E4:34_1_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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a></i>;<br/><a NAME="E5:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E8:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>;<br/><b>hence </b><a NAME="E6:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T32">GOBRD13:32</a></i>;<br/><a NAME="E7:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T39">GOBRD13:39</a></i>;<br/><a NAME="E8:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E8:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_3"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/><b>hence </b><a NAME="E9:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</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="34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><b>suppose </b></a><b>that </b><a NAME="E1:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>and </b><br/><a NAME="E2:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/><i><font color="Green">E67</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>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>and </b><br/><a NAME="E3:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/><i><font color="Green">E68</font></i>: 
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font>
 ;<br/><div class="add"><a NAME="E4:34_1_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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E63</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a></i>;<br/><a NAME="E5:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a></i>;<br/><b>hence </b><a NAME="E6:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="gobrd13.html#T32">GOBRD13:32</a></i>;<br/><a NAME="E7:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"/>
 <a href="gobrd13.html#K6">front_left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green">E66</font></i></a>, <a class="ref" href="gobrd13.html#T41">GOBRD13:41</a></i>;<br/><a NAME="E8:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1_1_1_3"><i><font color="Green">E64</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green">E67</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_1_1_1_3_1_1_4"><i><font color="Green">E68</font></i></a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/><b>hence </b><a NAME="E9:34_1_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_1"><i><font color="Green">E54</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="E4:34_1_1_2_1_1_3_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:34_1_1_2_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2_1_1_3_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_1_2_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>12</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><div class="add"><a NAME="E2:34_1_1_2_1_1_3_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>12</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><a NAME="E3:34_1_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E10:34_1_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2"><i><font color="Green">E47</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>13</sub></font>, <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>15</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E5:34_1_1_2_1_1_3_1_1_2"/><i><font color="Green">E55</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</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>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> )
 <b>and </b><br/><a NAME="E6:34_1_1_2_1_1_3_1_1_2"/><i><font color="Green">E56</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> )
 <b>and </b><br/><a NAME="E7:34_1_1_2_1_1_3_1_1_2"/><i><font color="Green">E57</font></i>: 
( ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:34_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="jordan8.html#T6">JORDAN8:6</a></i>;<br/><a NAME="E8:34_1_1_2_1_1_3_1_1_2"/><i><font color="Green">E58</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E9:34_1_1_2_1_1_3_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:34_1_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>12</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan9.html#E8:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E58</font></i></a>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E11:34_1_1_2_1_1_3_1_1_2"/><i><font color="Green">E60</font></i>: 
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:34_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E54</font></i></a></i>;<br/><a NAME="E12:34_1_1_2_1_1_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="finseq_1.html#K12">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="jordan9.html#E2:34_1_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="jordan8.html#T2">JORDAN8:2</a></i>;<br/><a NAME="E13:34_1_1_2_1_1_3_1_1_2"/><b>then </b><i><font color="Green">E61</font></i>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</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>12</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</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>12</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E5:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E6:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E10:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E59</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="34_1_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="34_1_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="E1:34_1_1_2_1_1_3_1_1_2_1_1"/>
( ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> ) or ( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E7:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E57</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2_1_1_3_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:34_1_1_2_1_1_3_1_1_2_1_1_1"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 ;<br/><div class="add"><a NAME="E2:34_1_1_2_1_1_3_1_1_2_1_1_1"/><b>then </b>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>14</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:34_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan9.html#E5:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E6:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="gobrd13.html#T22">GOBRD13:22</a>, <a class="ref" href="gobrd13.html#T23">GOBRD13:23</a></i>;<br/><b>hence </b><a NAME="E3:34_1_1_2_1_1_3_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E4:34_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E5:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E6:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan9.html#E13:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_2_1_1_1"><i><font color="Green">E62</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="34_1_1_2_1_1_3_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:34_1_1_2_1_1_3_1_1_2_1_1_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 ;<br/><div class="add"><a NAME="E2:34_1_1_2_1_1_3_1_1_2_1_1_2"/><b>then </b>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>14</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:34_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan9.html#E5:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E6:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="gobrd13.html#T24">GOBRD13:24</a>, <a class="ref" href="gobrd13.html#T25">GOBRD13:25</a></i>;<br/><b>hence </b><a NAME="E3:34_1_1_2_1_1_3_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E4:34_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E5:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E6:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan9.html#E13:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_2_1_1_2"><i><font color="Green">E62</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="34_1_1_2_1_1_3_1_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:34_1_1_2_1_1_3_1_1_2_1_1_3"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> )
 ;<br/><div class="add"><a NAME="E2:34_1_1_2_1_1_3_1_1_2_1_1_3"/><b>then </b>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>16</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:34_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan9.html#E5:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E6:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="gobrd13.html#T26">GOBRD13:26</a>, <a class="ref" href="gobrd13.html#T27">GOBRD13:27</a></i>;<br/><b>hence </b><a NAME="E3:34_1_1_2_1_1_3_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E4:34_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E5:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E6:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan9.html#E13:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_2_1_1_3"><i><font color="Green">E62</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="34_1_1_2_1_1_3_1_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:34_1_1_2_1_1_3_1_1_2_1_1_4"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:34_1_1_2_1_1_3_1_1_2_1_1_4"/><b>then </b>
(  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="finseq_1.html#K17">|</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="binarith.html#K5">-'</a> 1</span>)</span>,<font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E3:34_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E3:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan9.html#E5:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E6:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="ref" href="gobrd13.html#T28">GOBRD13:28</a>, <a class="ref" href="gobrd13.html#T29">GOBRD13:29</a></i>;<br/><b>hence </b><a NAME="E3:34_1_1_2_1_1_3_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E4:34_1_1_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan9.html#E15:34_1_1_2"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jordan9.html#E5:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan9.html#E6:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E56</font></i></a>, <a class="txt" href="jordan9.html#E11:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E60</font></i></a>, <a class="txt" href="jordan9.html#E13:34_1_1_2_1_1_3_1_1_2"><i><font color="Green">E61</font></i></a>, <a class="txt" href="jordan9.html#E1:34_1_1_2_1_1_3_1_1_2_1_1_4"><i><font color="Green">E62</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="E15:34_1_1_2_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:34_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E17:34_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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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>10</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">c<sub>12</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E6:34"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <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="jordan9.html#E3:34"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan9.html#E5:34"><i><font color="Green">E36</font></i></a>);<br/></i>
<a NAME="E7:34"/>
<span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</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="E8:34"/>
(  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> or  not <font color="Maroon">c<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"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> or (  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> &amp;  <a href="gobrd13.html#K3">right_cell</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1" target="_self">Cage</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">c<sub>3</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>3</sub></font> ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E6:34"><i><font color="Green">E37</font></i></a></i>;<br/>


</div>
