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

<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">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>;<br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="struct_0.html#NM7" title="STRUCT_0:NM.7">FinSequence</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:12"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E2:12"/><i><font color="Green" title="E14">A2</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">k</font>
 <b>and </b><br/><a NAME="E3:12"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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="c3">f</font>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">i1</font>, <font color="Maroon" title="c5">j1</font>, <font color="Maroon" title="c6">i2</font>, <font color="Maroon" title="c7">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="E5:12"/><i><font color="Green" title="E16">A4</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c4">i1</font>,<font color="Maroon" title="c5">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> <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E6:12"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i1</font>,<font color="Maroon" title="c5">j1</font>
 <b>and </b><br/><a NAME="E7:12"/><i><font color="Green" title="E18">A6</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">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> <font color="Maroon" title="c2">G</font>
 <b>and </b><br/><a NAME="E8:12"/><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">k</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>
 <b>and </b><br/><a NAME="E9:12"/><i><font color="Green" title="E20">A8</font></i>: 
( ( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</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="c7">j2</font> ) or ( <font color="Maroon" title="c4">i1</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="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> ) or ( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> ) or ( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 <b>by </b><i><a class="txt" href="jordan9.html#E1:12"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:12"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:12"><i><font color="Green" title="E15">A3</font></i></a>, <a class="ref" href="jordan8.html#T6" title="JORDAN8:th.6">JORDAN8:6</a></i>;<br/>
<a NAME="E10:12"/><i><font color="Green" title="E21">A9</font></i>: 
( <font color="Maroon" title="c4">i1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">i1</font> &amp; <font color="Maroon" title="c5">j1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c5">j1</font> &amp; <font color="Maroon" title="c6">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c7">j2</font> )
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E11:12"/><i><font color="Green" title="E22">A10</font></i>: 
( <font color="Maroon" title="c4">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> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c5">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> <font color="Maroon" title="c2">G</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E5:12"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E12:12"/><i><font color="Green" title="E23">A11</font></i>: 
( <font color="Maroon" title="c6">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> <font color="Maroon" title="c2">G</font> &amp; <font color="Maroon" title="c7">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> <font color="Maroon" title="c2">G</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E7:12"><i><font color="Green" title="E18">A6</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E13:12"/><i><font color="Green" title="E24">A12</font></i>: 
( <font color="Maroon" title="c4">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">G</font> &amp; <font color="Maroon" title="c5">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <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> <font color="Maroon" title="c2">G</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E11:12"><i><font color="Green" title="E22">A10</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/>
<a NAME="E14:12"/><i><font color="Green" title="E25">A13</font></i>: 
( <font color="Maroon" title="c6">i2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</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">G</font> &amp; <font color="Maroon" title="c7">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 <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> <font color="Maroon" title="c2">G</font> )
 
<b>by </b><i><a class="txt" href="jordan9.html#E12:12"><i><font color="Green" title="E23">A11</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:12_1"/>
( ( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</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="c7">j2</font> ) or ( <font color="Maroon" title="c4">i1</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="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> ) or ( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> ) or ( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E9:12"><i><font color="Green" title="E20">A8</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_1"><b>suppose </b></a><a NAME="E1:12_1_1"/><i><font color="Green" title="E26">A14</font></i>: 
( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</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="c7">j2</font> )
 ;<br/><div class="add"><a NAME="E2:12_1_1"/><b>then </b><i><font color="Green" title="E27">A15</font></i>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">i1</font>,<font color="Maroon" title="c5">j1</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:12"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:12"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:12"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="jordan9.html#E5:12"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="jordan9.html#E6:12"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="jordan9.html#E7:12"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="jordan9.html#E8:12"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="jordan9.html#E10:12"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></i>;<br/><a NAME="E3:12_1_1"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c2">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c5">j1</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:12"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:12"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:12"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="jordan9.html#E5:12"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="jordan9.html#E6:12"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="jordan9.html#E7:12"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="jordan9.html#E8:12"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="jordan9.html#E10:12"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="jordan9.html#E1:12_1_1"><i><font color="Green" title="E26">A14</font></i></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></i>;<br/><b>hence </b><a NAME="E4:12_1_1"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> )
 <b>by </b><i><a class="txt" href="jordan9.html#E11:12"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="jordan9.html#E13:12"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="jordan9.html#E2:12_1_1"><i><font color="Green" title="E27">A15</font></i></a>, <a class="ref" href="goboard9.html#T21" title="GOBOARD9:th.21">GOBOARD9:21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_2"><b>suppose </b></a><a NAME="E1:12_1_2"/><i><font color="Green" title="E26">A16</font></i>: 
( <font color="Maroon" title="c4">i1</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="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> )
 ;<br/><div class="add"><a NAME="E2:12_1_2"/><b>then </b><i><font color="Green" title="E27">A17</font></i>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">i1</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c5">j1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:12"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:12"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:12"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="jordan9.html#E5:12"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="jordan9.html#E6:12"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="jordan9.html#E7:12"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="jordan9.html#E8:12"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="jordan9.html#E10:12"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></i>;<br/><a NAME="E3:12_1_2"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">i1</font>,<font color="Maroon" title="c5">j1</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:12"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:12"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:12"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="jordan9.html#E5:12"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="jordan9.html#E6:12"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="jordan9.html#E7:12"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="jordan9.html#E8:12"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="jordan9.html#E10:12"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="jordan9.html#E1:12_1_2"><i><font color="Green" title="E26">A16</font></i></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></i>;<br/><b>hence </b><a NAME="E4:12_1_2"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> )
 <b>by </b><i><a class="txt" href="jordan9.html#E11:12"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="jordan9.html#E13:12"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="jordan9.html#E2:12_1_2"><i><font color="Green" title="E27">A17</font></i></a>, <a class="ref" href="goboard9.html#T21" title="GOBOARD9:th.21">GOBOARD9:21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_3"><b>suppose </b></a><a NAME="E1:12_1_3"/><i><font color="Green" title="E26">A18</font></i>: 
( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> )
 ;<br/><div class="add"><a NAME="E2:12_1_3"/><b>then </b><i><font color="Green" title="E27">A19</font></i>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c6">i2</font>,<font color="Maroon" title="c7">j2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:12"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:12"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:12"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="jordan9.html#E5:12"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="jordan9.html#E6:12"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="jordan9.html#E7:12"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="jordan9.html#E8:12"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="jordan9.html#E10:12"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></i>;<br/><a NAME="E3:12_1_3"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c6">i2</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c7">j2</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:12"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:12"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:12"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="jordan9.html#E5:12"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="jordan9.html#E6:12"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="jordan9.html#E7:12"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="jordan9.html#E8:12"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="jordan9.html#E10:12"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="jordan9.html#E1:12_1_3"><i><font color="Green" title="E26">A18</font></i></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></i>;<br/><b>hence </b><a NAME="E4:12_1_3"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> )
 <b>by </b><i><a class="txt" href="jordan9.html#E12:12"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="jordan9.html#E14:12"><i><font color="Green" title="E25">A13</font></i></a>, <a class="txt" href="jordan9.html#E2:12_1_3"><i><font color="Green" title="E27">A19</font></i></a>, <a class="ref" href="goboard9.html#T21" title="GOBOARD9:th.21">GOBOARD9:21</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="12_1_4"><b>suppose </b></a><a NAME="E1:12_1_4"/><i><font color="Green" title="E26">A20</font></i>: 
( <font color="Maroon" title="c4">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">i2</font> &amp; <font color="Maroon" title="c5">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">j2</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:12_1_4"/><b>then </b><i><font color="Green" title="E27">A21</font></i>: 
 <a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c2">G</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">i1</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span>,<font color="Maroon" title="c7">j2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:12"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:12"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:12"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="jordan9.html#E5:12"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="jordan9.html#E6:12"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="jordan9.html#E7:12"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="jordan9.html#E8:12"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="jordan9.html#E10:12"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="gobrd13.html#D2" title="GOBRD13:def.2">GOBRD13:def 2</a></i>;<br/><a NAME="E3:12_1_4"/>
 <a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c2">G</font>,<font color="Maroon" title="c4">i1</font>,<font color="Maroon" title="c7">j2</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E1:12"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="jordan9.html#E2:12"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="jordan9.html#E3:12"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="jordan9.html#E5:12"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="jordan9.html#E6:12"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="jordan9.html#E7:12"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="jordan9.html#E8:12"><i><font color="Green" title="E19">A7</font></i></a>, <a class="txt" href="jordan9.html#E10:12"><i><font color="Green" title="E21">A9</font></i></a>, <a class="txt" href="jordan9.html#E1:12_1_4"><i><font color="Green" title="E26">A20</font></i></a>, <a class="ref" href="gobrd13.html#D3" title="GOBRD13:def.3">GOBRD13:def 3</a></i>;<br/><b>hence </b><a NAME="E4:12_1_4"/>
(  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4" title="GOBRD13:func.4">left_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> &amp;  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K3" title="GOBRD13:func.3">right_cell</a> <font color="Maroon" title="c3">f</font>,<font color="Maroon" title="c1">k</font>,<font color="Maroon" title="c2">G</font></span>)</span> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> )
 <b>by </b><i><a class="txt" href="jordan9.html#E11:12"><i><font color="Green" title="E22">A10</font></i></a>, <a class="txt" href="jordan9.html#E12:12"><i><font color="Green" title="E23">A11</font></i></a>, <a class="txt" href="jordan9.html#E13:12"><i><font color="Green" title="E24">A12</font></i></a>, <a class="txt" href="jordan9.html#E2:12_1_4"><i><font color="Green" title="E27">A21</font></i></a>, <a class="ref" href="goboard9.html#T21" title="GOBOARD9:th.21">GOBOARD9:21</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
