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<b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be  <a href="goboard5.html#V2" title="GOBOARD5:attr.2">standard</a> <a href="goboard5.html#NM1" title="GOBOARD5:NM.1">special_circular_sequence</a>;<br/>



<b>assume </b><a NAME="E1:41"/><i><font color="Green" title="E39">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> &amp; <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font> )
 ;<br/>

<a NAME="E2:41"/>
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E3:41"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:41"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:41"/><b>then </b><i><font color="Green" title="E40">A2</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:41"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">i1</font>, <font color="Maroon" title="c4">j1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:41"/><i><font color="Green" title="E41">A3</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">j1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">j1</font> )
 <b>by </b><i><a class="ref" href="goboard5.html#T12" target="_self" title="GOBOARD5:th.12">Th12</a></i>;<br/>
<a NAME="E7:41"/>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></i>;<br/>
<a NAME="E8:41"/><b>then </b><i><font color="Green" title="E42">A4</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">f</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E1:41"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c5">i2</font>, <font color="Maroon" title="c6">j2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:41"/><i><font color="Green" title="E43">A5</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c5">i2</font>,<font color="Maroon" title="c6">j2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="matrix_1.html#K2" title="MATRIX_1:func.2">Indices</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> &amp; <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i2</font>,<font color="Maroon" title="c6">j2</font> )
 <b>by </b><i><a class="ref" href="goboard5.html#T12" target="_self" title="GOBOARD5:th.12">Th12</a></i>;<br/>
<a NAME="E11:41"/><i><font color="Green" title="E44">A6</font></i>: 
<span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">i2</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="txt" href="goboard5.html#E4:41"><i><font color="Green" title="E40">A2</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="goboard5.html#E8:41"><i><font color="Green" title="E42">A4</font></i></a>, <a class="txt" href="goboard5.html#E10:41"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="goboard5.html#T13" target="_self" title="GOBOARD5:th.13">Th13</a></i>;<br/>
<div><i><font color="Green" title="E45">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1"/>
( (  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> ) or ( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> &amp;  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E11:41"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="goboard1.html#T2" title="GOBOARD1:th.2">GOBOARD1:2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1"><b>case </b></a><b>that </b><a NAME="E1:41_1_1_1"/><i><font color="Green" title="E46">A8</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>and </b><br/><a NAME="E2:41_1_1_1"/><i><font color="Green" title="E47">A9</font></i>: 
<font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font>
 ;<br/><div class="add"><a NAME="E3:41_1_1_1"/>
( <font color="Maroon" title="c3">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">i2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c3">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><a NAME="E4:41_1_1_1"/><b>then </b>
( <font color="Maroon" title="c3">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">i2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:41_1_1_1"><i><font color="Green" title="E46">A8</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:41_1_1_1"/>
( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c3">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> )
 ;<br/><b>thus </b><a NAME="E6:41_1_1_1"/>
<font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font>
 <b>by </b><i><a class="txt" href="goboard5.html#E2:41_1_1_1"><i><font color="Green" title="E47">A9</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2"><b>case </b></a><b>that </b><a NAME="E1:41_1_1_2"/><i><font color="Green" title="E46">A10</font></i>: 
<font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font>
 <b>and </b><br/><a NAME="E2:41_1_1_2"/><i><font color="Green" title="E47">A11</font></i>: 
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E3:41_1_1_2"/>
( <font color="Maroon" title="c4">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 ;<br/><a NAME="E4:41_1_1_2"/><b>then </b>
( <font color="Maroon" title="c4">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or  <a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">j1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c6">j2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="goboard5.html#E2:41_1_1_2"><i><font color="Green" title="E47">A11</font></i></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</a></i>;<br/><b>hence </b><a NAME="E5:41_1_1_2"/>
( <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c4">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> )
 ;<br/><b>thus </b><a NAME="E6:41_1_1_2"/>
<font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font>
 <b>by </b><i><a class="txt" href="goboard5.html#E1:41_1_1_2"><i><font color="Green" title="E46">A10</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E13:41"/><i><font color="Green" title="E46">A12</font></i>: 
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j1</font> &amp; <font color="Maroon" title="c4">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard5.html#E6:41"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="goboard5.html#T1" target="_self" title="GOBOARD5:th.1">Th1</a></i>;<br/>
<a NAME="E14:41"/><i><font color="Green" title="E47">A13</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">j2</font> &amp; <font color="Maroon" title="c6">j2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard5.html#E10:41"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="goboard5.html#T1" target="_self" title="GOBOARD5:th.1">Th1</a></i>;<br/>
<a NAME="E15:41"/><i><font color="Green" title="E48">A14</font></i>: 
( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i1</font> &amp; <font color="Maroon" title="c3">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard5.html#E6:41"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="goboard5.html#T1" target="_self" title="GOBOARD5:th.1">Th1</a></i>;<br/>
<a NAME="E16:41"/><i><font color="Green" title="E49">A15</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c5">i2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="goboard5.html#E10:41"><i><font color="Green" title="E43">A5</font></i></a>, <a class="ref" href="goboard5.html#T1" target="_self" title="GOBOARD5:th.1">Th1</a></i>;<br/>
<a NAME="E17:41"/>
<font color="Maroon" title="c3">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="goboard5.html#E15:41"><i><font color="Green" title="E48">A14</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E19:41"/><i><font color="Green" title="E50">A16</font></i>: 
<font color="Maroon" title="c7">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">i</font> = <font color="Maroon" title="c7">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E21:41"/><i><font color="Green" title="E51">A17</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i1</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E19:41"><i><font color="Green" title="E50">A16</font></i></a></i>;<br/>
<a NAME="E22:41"/><i><font color="Green" title="E52">A18</font></i>: 
<font color="Maroon" title="c8">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E15:41"><i><font color="Green" title="E48">A14</font></i></a>, <a class="txt" href="goboard5.html#E19:41"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E23:41"/>
<font color="Maroon" title="c4">j1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="txt" href="goboard5.html#E13:41"><i><font color="Green" title="E46">A12</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">j</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E25:41"/><i><font color="Green" title="E53">A19</font></i>: 
<font color="Maroon" title="c9">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T6" title="NAT_1:th.6">NAT_1:6</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">j</font> = <font color="Maroon" title="c9">j</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E27:41"/><i><font color="Green" title="E54">A20</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j1</font>
 
<b>by </b><i><a class="txt" href="goboard5.html#E25:41"><i><font color="Green" title="E53">A19</font></i></a></i>;<br/>
<a NAME="E28:41"/><i><font color="Green" title="E55">A21</font></i>: 
<font color="Maroon" title="c10">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 
<b>by </b><i><a class="txt" href="goboard5.html#E13:41"><i><font color="Green" title="E46">A12</font></i></a>, <a class="txt" href="goboard5.html#E25:41"><i><font color="Green" title="E53">A19</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_2"/>
( ( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c4">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> ) or ( <font color="Maroon" title="c3">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> ) or ( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> ) or ( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="goboard5.html#E12:41"><i><font color="Green" title="E45">A7</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_1"><b>suppose </b></a><a NAME="E1:41_2_1"/><i><font color="Green" title="E56">A22</font></i>: 
( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c4">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> )
 ;<br/><div class="add"><a NAME="E2:41_2_1"/><b>then </b><i><font color="Green" title="E57">A23</font></i>: 
<font color="Maroon" title="c4">j1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E14:41"><i><font color="Green" title="E47">A13</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:41_2_1"/>
(  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c4">j1</font> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">j1</font> )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:41"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="goboard5.html#E10:41"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="goboard5.html#E21:41"><i><font color="Green" title="E51">A17</font></i></a>, <a class="txt" href="goboard5.html#E1:41_2_1"><i><font color="Green" title="E56">A22</font></i></a>, <a class="ref" href="goboard5.html#T28" target="_self" title="GOBOARD5:th.28">Th28</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font></span>)</span> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">j1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">j1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="goboard5.html#E13:41"><i><font color="Green" title="E46">A12</font></i></a>, <a class="txt" href="goboard5.html#E19:41"><i><font color="Green" title="E50">A16</font></i></a>, <a class="txt" href="goboard5.html#E22:41"><i><font color="Green" title="E52">A18</font></i></a>, <a class="txt" href="goboard5.html#E2:41_2_1"><i><font color="Green" title="E57">A23</font></i></a>, <a class="ref" href="goboard5.html#T26" target="_self" title="GOBOARD5:th.26">Th26</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font>
<b>by </b><i><a class="txt" href="goboard5.html#E1:41"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="goboard5.html#E10:41"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="goboard5.html#E1:41_2_1"><i><font color="Green" title="E56">A22</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_2"><b>suppose </b></a><a NAME="E1:41_2_2"/><i><font color="Green" title="E56">A24</font></i>: 
( <font color="Maroon" title="c3">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> )
 ;<br/><div class="add"><a NAME="E2:41_2_2"/><b>then </b><i><font color="Green" title="E57">A25</font></i>: 
<font color="Maroon" title="c3">i1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E16:41"><i><font color="Green" title="E49">A15</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:41_2_2"/>
(  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">j1</font> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c10">j</font> )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:41"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="goboard5.html#E10:41"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="goboard5.html#E27:41"><i><font color="Green" title="E54">A20</font></i></a>, <a class="txt" href="goboard5.html#E1:41_2_2"><i><font color="Green" title="E56">A24</font></i></a>, <a class="ref" href="goboard5.html#T29" target="_self" title="GOBOARD5:th.29">Th29</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font></span>)</span> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c4">j1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">j1</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard5.html#E15:41"><i><font color="Green" title="E48">A14</font></i></a>, <a class="txt" href="goboard5.html#E25:41"><i><font color="Green" title="E53">A19</font></i></a>, <a class="txt" href="goboard5.html#E28:41"><i><font color="Green" title="E55">A21</font></i></a>, <a class="txt" href="goboard5.html#E2:41_2_2"><i><font color="Green" title="E57">A25</font></i></a>, <a class="ref" href="goboard5.html#T27" target="_self" title="GOBOARD5:th.27">Th27</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font>
<b>by </b><i><a class="txt" href="goboard5.html#E1:41"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="goboard5.html#E10:41"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="goboard5.html#E1:41_2_2"><i><font color="Green" title="E56">A24</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_3"><b>suppose </b></a><a NAME="E1:41_2_3"/><i><font color="Green" title="E56">A26</font></i>: 
( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> )
 ;<br/><div class="add"><a NAME="E2:41_2_3"/><b>then </b><i><font color="Green" title="E57">A27</font></i>: 
<font color="Maroon" title="c5">i2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E15:41"><i><font color="Green" title="E48">A14</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:41_2_3"/>
(  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c5">i2</font>,<font color="Maroon" title="c10">j</font> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c5">i2</font>,<font color="Maroon" title="c4">j1</font> )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:41"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="goboard5.html#E10:41"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="goboard5.html#E27:41"><i><font color="Green" title="E54">A20</font></i></a>, <a class="txt" href="goboard5.html#E1:41_2_3"><i><font color="Green" title="E56">A26</font></i></a>, <a class="ref" href="goboard5.html#T30" target="_self" title="GOBOARD5:th.30">Th30</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font></span>)</span> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c5">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c4">j1</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c5">i2</font>,<font color="Maroon" title="c4">j1</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard5.html#E16:41"><i><font color="Green" title="E49">A15</font></i></a>, <a class="txt" href="goboard5.html#E25:41"><i><font color="Green" title="E53">A19</font></i></a>, <a class="txt" href="goboard5.html#E28:41"><i><font color="Green" title="E55">A21</font></i></a>, <a class="txt" href="goboard5.html#E2:41_2_3"><i><font color="Green" title="E57">A27</font></i></a>, <a class="ref" href="goboard5.html#T27" target="_self" title="GOBOARD5:th.27">Th27</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font>
<b>by </b><i><a class="txt" href="goboard5.html#E1:41"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="goboard5.html#E10:41"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="goboard5.html#E1:41_2_3"><i><font color="Green" title="E56">A26</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_2_4"><b>suppose </b></a><a NAME="E1:41_2_4"/><i><font color="Green" title="E56">A28</font></i>: 
( <font color="Maroon" title="c3">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">i2</font> &amp; <font color="Maroon" title="c4">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:41_2_4"/><b>then </b><i><font color="Green" title="E57">A29</font></i>: 
<font color="Maroon" title="c6">j2</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard5.html#E13:41"><i><font color="Green" title="E46">A12</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:41_2_4"/>
(  <a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c6">j2</font> &amp;  <a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span>,<font color="Maroon" title="c8">i</font>,<font color="Maroon" title="c6">j2</font> )
 <b>by </b><i><a class="txt" href="goboard5.html#E1:41"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="goboard5.html#E10:41"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="goboard5.html#E21:41"><i><font color="Green" title="E51">A17</font></i></a>, <a class="txt" href="goboard5.html#E1:41_2_4"><i><font color="Green" title="E56">A28</font></i></a>, <a class="ref" href="goboard5.html#T31" target="_self" title="GOBOARD5:th.31">Th31</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="goboard5.html#K5" title="GOBOARD5:func.5">left_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="goboard5.html#K4" title="GOBOARD5:func.4">right_cell</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font></span>)</span> = 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i1</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c6">j2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="goboard2.html#K3" title="GOBOARD2:func.3">GoB</a> <font color="Maroon" title="c2">f</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i1</font>,<font color="Maroon" title="c6">j2</font></span>)</span>
<b>by </b><i><a class="txt" href="goboard5.html#E14:41"><i><font color="Green" title="E47">A13</font></i></a>, <a class="txt" href="goboard5.html#E19:41"><i><font color="Green" title="E50">A16</font></i></a>, <a class="txt" href="goboard5.html#E22:41"><i><font color="Green" title="E52">A18</font></i></a>, <a class="txt" href="goboard5.html#E2:41_2_4"><i><font color="Green" title="E57">A29</font></i></a>, <a class="ref" href="goboard5.html#T26" target="_self" title="GOBOARD5:th.26">Th26</a></i>
<br/>.= 
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c2">f</font>,<font color="Maroon" title="c1">k</font>
<b>by </b><i><a class="txt" href="goboard5.html#E1:41"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="goboard5.html#E6:41"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="goboard5.html#E10:41"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="goboard5.html#E1:41_2_4"><i><font color="Green" title="E56">A28</font></i></a>, <a class="ref" href="topreal1.html#D5" title="TOPREAL1:def.5">TOPREAL1:def 5</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div>

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