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<b>let </b><font color="Maroon" title="c1">i1</font>, <font color="Maroon" title="c2">j1</font>, <font color="Maroon" title="c3">i2</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="c4">G1</font>, <font color="Maroon" title="c5">G2</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:16"/><i><font color="Green" title="E10">A1</font></i>: 
 <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c4">G1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c5">G2</font>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green" title="E11">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">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="c4">G1</font>
 <b>and </b><br/><a NAME="E3:16"/><i><font color="Green" title="E12">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i2</font> &amp; <font color="Maroon" title="c3">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="c5">G2</font> )
 <b>and </b><br/><a NAME="E4:16"/><i><font color="Green" title="E13">A4</font></i>: 
<font color="Maroon" title="c4">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i2</font>,1
 ;<br/>

<a NAME="E5:16"/><i><font color="Green" title="E14">A5</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">i1</font> &amp; <font color="Maroon" title="c1">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="c4">G1</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">j1</font> &amp; <font color="Maroon" title="c2">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="c4">G1</font> )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E2:16"><i><font color="Green" title="E11">A2</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E6:16"/>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c5">G2</font>
 
<b>by </b><i><a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a></i>;<br/>
<a NAME="E7:16"/><b>then </b><i><font color="Green" title="E15">A6</font></i>: 
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="c5">G2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E8:16"/>
 <a href="card_1.html#NK1" title="CARD_1:NK.1">0</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c4">G1</font>
 
<b>by </b><i><a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a></i>;<br/>
<a NAME="E9:16"/><b>then </b><i><font color="Green" title="E16">A7</font></i>: 
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="c4">G1</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<b>set </b><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c4">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,1;<br/>
<b>assume </b><a NAME="E10:16"/>
<font color="Maroon" title="c2">j1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ;<br/>

<a NAME="E11:16"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j1</font>
 
<b>by </b><i><a class="txt" href="gobrd13.html#E5:16"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E12:16"/><b>then </b><i><font color="Green" title="E17">A8</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c2">j1</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 
<b>by </b><i><a class="txt" href="gobrd13.html#E5:16"><i><font color="Green" title="E14">A5</font></i></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></i>;<br/>
<a NAME="E13:16"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">i1</font>,1</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="c4">G1</font>
 
<b>by </b><i><a class="txt" href="gobrd13.html#E5:16"><i><font color="Green" title="E14">A5</font></i></a>, <a class="txt" href="gobrd13.html#E9:16"><i><font color="Green" title="E16">A7</font></i></a>, <a class="ref" href="goboard7.html#T10" title="GOBOARD7:th.10">GOBOARD7:10</a></i>;<br/>
<a NAME="E14:16"/><b>then </b>
<font color="Maroon" title="c4">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c4">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <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>  : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</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="c4">G1</font> </span>}</span> 
 
;<br/>
<a NAME="E15:16"/><b>then </b>
<font color="Maroon" title="c4">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c4">G1</font>
 
<b>by </b><i><a class="ref" href="gobrd13.html#T7" target="_self" title="GOBRD13:th.7">Th7</a></i>;<br/>
<a NAME="E16:16"/><b>then </b>
<font color="Maroon" title="c4">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="gobrd13.html#K2" title="GOBRD13:func.2">Values</a> <font color="Maroon" title="c5">G2</font>
 
<b>by </b><i><a class="txt" href="gobrd13.html#E1:16"><i><font color="Green" title="E10">A1</font></i></a></i>;<br/>
<a NAME="E17:16"/><b>then </b>
<font color="Maroon" title="c4">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c5">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</font></span>)</span> where <font color="Olive" title="b1">i</font>, <font color="Olive" title="b2">j</font> is    <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>  : <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Olive" title="b1">i</font>,<font color="Olive" title="b2">j</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="c5">G2</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="gobrd13.html#T7" target="_self" title="GOBRD13:th.7">Th7</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">j</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="E19:16"/><i><font color="Green" title="E18">A9</font></i>: 
<font color="Maroon" title="c4">G1</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</font>
 <b>and </b><br/><a NAME="E20:16"/><i><font color="Green" title="E19">A10</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c7">i</font>,<font color="Maroon" title="c8">j</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="c5">G2</font>
 ;<br/>
<a NAME="E21:16"/><i><font color="Green" title="E20">A11</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">i</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="c5">G2</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</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="c5">G2</font> )
 
<b>by </b><i><a class="txt" href="gobrd13.html#E20:16"><i><font color="Green" title="E19">A10</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<b>then </b><i><font color="Green" title="E21">A12</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c7">i</font>,1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
<b>by </b><i><a class="txt" href="gobrd13.html#E7:16"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G2</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i2</font>,1</span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
<b>by </b><i><a class="txt" href="gobrd13.html#E3:16"><i><font color="Green" title="E12">A3</font></i></a>, <a class="txt" href="gobrd13.html#E7:16"><i><font color="Green" title="E15">A6</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>
;<br/>
<a NAME="E23:16"/><b>then </b>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c8">j</font>
 
<b>by </b><i><a class="txt" href="gobrd13.html#E4:16"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="gobrd13.html#E12:16"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="gobrd13.html#E19:16"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="gobrd13.html#E21:16"><i><font color="Green" title="E20">A11</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E24:16"/>
contradiction
 <b>by </b><i><a class="txt" href="gobrd13.html#E4:16"><i><font color="Green" title="E13">A4</font></i></a>, <a class="txt" href="gobrd13.html#E12:16"><i><font color="Green" title="E17">A8</font></i></a>, <a class="txt" href="gobrd13.html#E19:16"><i><font color="Green" title="E18">A9</font></i></a>, <a class="txt" href="gobrd13.html#E21:16"><i><font color="Green" title="E20">A11</font></i></a>, <a class="txt" href="gobrd13.html#E22:16"><i><font color="Green" title="E21">A12</font></i></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></i>;<br/>


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