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

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">f</font> be  <a href="goboard5.html#V2">standard</a> <a href="goboard5.html#NM1">special_circular_sequence</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:34"/><i><font color="Green">E59</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 ;<br/>

<b>set </b><font color="Maroon">G</font> =  <a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font>;<br/>
<a NAME="E3:34"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">f</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="goboard5.html#D5">GOBOARD5:def 5</a></i>;<br/>
<a NAME="E4:34"/>
for <font color="Olive">i1</font>, <font color="Olive">j1</font>, <font color="Olive">i2</font>, <font color="Olive">j2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i1</font>,<font color="Olive">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">i2</font>,<font color="Olive">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i1</font>,<font color="Olive">j1</font> &amp; <font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">i2</font>,<font color="Olive">j2</font> &amp; not <!--HUMANRECFAILED-->( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> &amp;  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Olive">j1</font> ) &amp; not <!--HUMANRECFAILED-->( <font color="Olive">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> &amp;  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<font color="Olive">i1</font>,<font color="Olive">j1</font> ) &amp; not <!--HUMANRECFAILED-->( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> &amp;  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<font color="Olive">i2</font>,<span class="p1">(<span class="default"><font color="Olive">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) holds <br/>( <font color="Olive">i1</font> <a href="hidden.html#R1">=</a> <font color="Olive">i2</font> &amp; <font color="Olive">j1</font> <a href="hidden.html#R1">=</a> <font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<font color="Olive">i1</font>,<font color="Olive">j2</font> )
 
<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">i1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">i2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:34_1"/><i><font color="Green">E68</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i1</font>,<font color="Maroon">j1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E2:34_1"/><i><font color="Green">E69</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i2</font>,<font color="Maroon">j2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>
 <b>and </b><br/><a NAME="E3:34_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i1</font>,<font color="Maroon">j1</font>
 <b>and </b><br/><a NAME="E4:34_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i2</font>,<font color="Maroon">j2</font>
 ;<br/>

<a NAME="E5:34_1"/>
 <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>
 
;<br/>
<a NAME="E6:34_1"/><b>then </b><i><font color="Green">E73</font></i>: 
( ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp;  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp;  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> &amp;  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j2</font> ) )
 
<b>by </b><i>, , , , , , <a class="ref" href="goboard5.html#D7">GOBOARD5:def 7</a></i>;<br/>
<a NAME="E7:34_1"/>
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<a NAME="E8:34_1"/><b>then </b><i><font color="Green">E74</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E9:34_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E10:34_1"/><b>then </b>
<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<a NAME="E11:34_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, , , , , , <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/>
<a NAME="E12:34_1"/><b>then </b><i><font color="Green">E75</font></i>: 
( (  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i2</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or (  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j1</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j2</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> ) )
 
<b>by </b><i><a class="ref" href="goboard1.html#T2">GOBOARD1:2</a></i>;<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">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> ) or ( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i>, <a class="ref" href="goboard1.html#T1">GOBOARD1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_1"><b>case </b></a><a NAME="E1:34_1_1_1"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_1_1_1"/>
 <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_2"><b>case </b></a><a NAME="E1:34_1_1_2"/>
( <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_1_1_2"/>
 <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j1</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_3"><b>case </b></a><a NAME="E1:34_1_1_3"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_1_1_3"/>
 <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">i2</font>,<span class="p1">(<span class="default"><font color="Maroon">j2</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="34_1_1_4"><b>case </b></a><a NAME="E1:34_1_1_4"/>
( <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">j1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:34_1_1_4"/>
 <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span>,<font color="Maroon">i1</font>,<font color="Maroon">j2</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>hence </b><a NAME="E5:34"/>
 <a href="gobrd13.html#K4">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>,<span class="p1">(<span class="default"><a href="goboard2.html#K3">GoB</a> <font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K5">left_cell</a> <font color="Maroon">f</font>,<font color="Maroon">k</font>
 <b>by </b><i>, , , <a class="ref" href="gobrd13.html#D3">GOBRD13:def 3</a></i>;<br/>


</div>
