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<b>let </b><font color="Maroon" title="c1">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>;<br/><b>let </b><font color="Maroon" title="c2">i1</font>, <font color="Maroon" title="c3">j1</font>, <font color="Maroon" title="c4">i2</font>, <font color="Maroon" title="c5">j2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:24"/><i><font color="Green" title="E21">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i1</font> &amp; <font color="Maroon" title="c2">i1</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="c1">G</font> )
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green" title="E22">A2</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j1</font> &amp; <font color="Maroon" title="c3">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="c1">G</font> )
 <b>and </b><br/><a NAME="E3:24"/><i><font color="Green" title="E23">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">i2</font> &amp; <font color="Maroon" title="c4">i2</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="c1">G</font> )
 <b>and </b><br/><a NAME="E4:24"/><i><font color="Green" title="E24">A4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">j2</font> &amp; <font color="Maroon" title="c5">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="c1">G</font> )
 <b>and </b><br/><a NAME="E5:24"/><i><font color="Green" title="E25">A5</font></i>: 
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">j1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">j1</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c5">j2</font></span>)</span>
 ;<br/>

<a NAME="E6:24"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c3">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">j2</font>
 
<b>by </b><i><a class="txt" href="goboard7.html#E1:24"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="goboard7.html#E2:24"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="goboard7.html#E3:24"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="goboard7.html#E4:24"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="goboard7.html#E5:24"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="goboard7.html#T22" target="_self" title="GOBOARD7:th.22">Th22</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">m</font> =  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">i2</font></span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E8:24"/>
<font color="Maroon" title="c6">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="goboard7.html#E1:24"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="goboard7.html#E2:24"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="goboard7.html#E3:24"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="goboard7.html#E4:24"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="goboard7.html#E5:24"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="goboard7.html#T22" target="_self" title="GOBOARD7:th.22">Th22</a></i>;<br/>
<a NAME="E9:24"/><b>then </b><i><font color="Green" title="E27">A7</font></i>: 
(  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">i1</font> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c4">i2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 
<b>by </b><i><a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/>
<a NAME="E10:24"/><i><font color="Green" title="E28">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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="c2">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
;<br/>
<a NAME="E11:24"/><i><font color="Green" title="E29">A9</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <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">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 
;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:24_1"/>
( <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font> or <font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 or <font color="Maroon" title="c2">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="c4">i2</font> )
 </span><b>by </b><i><a class="txt" href="goboard7.html#E9:24"><i><font color="Green" title="E27">A7</font></i></a>, <a class="ref" href="goboard7.html#T2" target="_self" title="GOBOARD7:th.2">Th2</a>, <a class="ref" href="goboard1.html#T1" title="GOBOARD1:th.1">GOBOARD1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>case </b></a><a NAME="E1:24_1_1"/>
<font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_1"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">j1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">j1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c5">j2</font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard7.html#E6:24"><i><font color="Green" title="E26">A6</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2"><b>case </b></a><a NAME="E1:24_1_2"/>
<font color="Maroon" title="c2">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_2"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">j1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">j1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c5">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">j1</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard7.html#E1:24"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="goboard7.html#E2:24"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="goboard7.html#E3:24"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="goboard7.html#E6:24"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="goboard7.html#E11:24"><i><font color="Green" title="E29">A9</font></i></a>, <a class="ref" href="goboard7.html#T16" target="_self" title="GOBOARD7:th.16">Th16</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_3"><b>case </b></a><a NAME="E1:24_1_3"/>
<font color="Maroon" title="c2">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="c4">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_3"/>
<span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i1</font>,<font color="Maroon" title="c3">j1</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">i1</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c3">j1</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i2</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c5">j2</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c4">i2</font>,<font color="Maroon" title="c5">j2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard7.html#E1:24"><i><font color="Green" title="E21">A1</font></i></a>, <a class="txt" href="goboard7.html#E2:24"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="goboard7.html#E3:24"><i><font color="Green" title="E23">A3</font></i></a>, <a class="txt" href="goboard7.html#E6:24"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="goboard7.html#E10:24"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="goboard7.html#T16" target="_self" title="GOBOARD7:th.16">Th16</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
