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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>, <font color="Maroon" title="c4">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>let </b><font color="Maroon" title="c5">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:39"/><i><font color="Green" title="E26">A1</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="c5">G</font>
 <b>and </b><br/><a NAME="E2:39"/><i><font color="Green" title="E27">A2</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c4">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="c5">G</font>
 <b>and </b><br/><a NAME="E3:39"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c5">G</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">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c3">i2</font>,<font color="Maroon" title="c4">j2</font>
 ;<br/>

<a NAME="E4:39"/><i><font color="Green" title="E29">A4</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="c5">G</font> )
 
<b>by </b><i><a class="txt" href="jordan1h.html#E1:39"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E5:39"/><i><font color="Green" title="E30">A5</font></i>: 
( 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="c5">G</font> )
 
<b>by </b><i><a class="txt" href="jordan1h.html#E1:39"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E6:39"/><i><font color="Green" title="E31">A6</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">G</font> )
 
<b>by </b><i><a class="txt" href="jordan1h.html#E2:39"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="goboard5.html#T1" title="GOBOARD5:th.1">GOBOARD5:1</a></i>;<br/>
<a NAME="E7:39"/><i><font color="Green" title="E32">A7</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">j2</font> &amp; <font color="Maroon" title="c4">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="c5">G</font> )
 
<b>by </b><i><a class="txt" href="jordan1h.html#E2:39"><i><font color="Green" title="E27">A2</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="E33">A8</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c4">j2</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</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#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="txt" href="jordan1h.html#E4:39"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</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#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="txt" href="jordan1h.html#E4:39"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="jordan1h.html#E5:39"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>
;<br/>
<i><font color="Green" title="E34">A9</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i1</font>,<font color="Maroon" title="c4">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c4">j2</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
<b>by </b><i><a class="txt" href="jordan1h.html#E4:39"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="jordan1h.html#E7:39"><i><font color="Green" title="E32">A7</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">G</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="jordan1h.html#E3:39"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="jordan1h.html#E6:39"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="jordan1h.html#E7:39"><i><font color="Green" title="E32">A7</font></i></a>, <a class="ref" href="goboard5.html#T2" title="GOBOARD5:th.2">GOBOARD5:2</a></i>
;<br/>
<b>assume </b><a NAME="E10:39"/><i><font color="Green" title="E35">A10</font></i>: 
(  not <font color="Maroon" title="c1">i1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">i2</font> or  not <font color="Maroon" title="c2">j1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">j2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_3"/>
( <font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">i2</font> or <font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">i2</font> or <font color="Maroon" title="c2">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">j2</font> or <font color="Maroon" title="c2">j1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">j2</font> )
 </span><b>by </b><i><a class="txt" href="jordan1h.html#E10:39"><i><font color="Green" title="E35">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_3_1"><b>suppose </b></a><a NAME="E1:39_3_1"/>
<font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c3">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_3_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1h.html#E3:39"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="jordan1h.html#E4:39"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="jordan1h.html#E6:39"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="jordan1h.html#E7:39"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="jordan1h.html#E8:39"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_3_2"><b>suppose </b></a><a NAME="E1:39_3_2"/>
<font color="Maroon" title="c1">i1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">i2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_3_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1h.html#E3:39"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="jordan1h.html#E4:39"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="jordan1h.html#E6:39"><i><font color="Green" title="E31">A6</font></i></a>, <a class="txt" href="jordan1h.html#E7:39"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="jordan1h.html#E8:39"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_3_3"><b>suppose </b></a><a NAME="E1:39_3_3"/>
<font color="Maroon" title="c2">j1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c4">j2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_3_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1h.html#E4:39"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="jordan1h.html#E5:39"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="jordan1h.html#E7:39"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="jordan1h.html#E9:39"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_3_4"><b>suppose </b></a><a NAME="E1:39_3_4"/>
<font color="Maroon" title="c2">j1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">j2</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:39_3_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1h.html#E4:39"><i><font color="Green" title="E29">A4</font></i></a>, <a class="txt" href="jordan1h.html#E5:39"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="jordan1h.html#E7:39"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="jordan1h.html#E9:39"><i><font color="Green" title="E34">A9</font></i></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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