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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">i</font>, <font color="Maroon" title="c3">j</font>, <font color="Maroon" title="c4">m</font>, <font color="Maroon" title="c5">n</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="c6">p</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</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="E1:24"/><i><font color="Green" title="E12">A1</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> &amp; <font color="Maroon" title="c2">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="c1">G</font> )
 <b>and </b><br/><a NAME="E2:24"/><i><font color="Green" title="E13">A2</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">j</font> &amp; <font color="Maroon" title="c3">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="c1">G</font> )
 <b>and </b><br/><a NAME="E3:24"/><i><font color="Green" title="E14">A3</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c4">m</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="c1">G</font> )
 <b>and </b><br/><a NAME="E4:24"/><i><font color="Green" title="E15">A4</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c5">n</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="E16">A5</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font>
 <b>and </b><br/><a NAME="E6:24"/><i><font color="Green" title="E17">A6</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">m</font>,<font color="Maroon" title="c5">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ;<br/>

<a NAME="E7:24"/><i><font color="Green" title="E18">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="c1">G</font>
 
<b>by </b><i><a class="txt" href="jordan1b.html#E2:24"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:24"/><i><font color="Green" title="E19">A8</font></i>: 
<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">m</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">m</font>,<font color="Maroon" title="c5">n</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan1b.html#E3:24"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="jordan1b.html#E4:24"><i><font color="Green" title="E15">A4</font></i></a>, <a class="ref" href="goboard5.html#T3" title="GOBOARD5:th.3">GOBOARD5:3</a></i>;<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">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font> or <font color="Maroon" title="c2">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="c1">G</font> )
 </span><b>by </b><i><a class="txt" href="jordan1b.html#E1:24"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_1"><b>suppose </b></a><a NAME="E1:24_1_1"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c1">G</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_1"/>
( <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> or <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 )
 <b>by </b><i><a class="txt" href="jordan1b.html#E2:24"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="jordan1b.html#E3:24"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="jordan1b.html#E4:24"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="jordan1b.html#E5:24"><i><font color="Green" title="E16">A5</font></i></a>, <a class="txt" href="jordan1b.html#E6:24"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="jordan1b.html#T22" target="_self" title="JORDAN1B:th.22">Th22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2"><b>suppose </b></a><a NAME="E1:24_1_2"/>
<font color="Maroon" title="c2">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="c1">G</font>
 ;<br/><div class="add"><a NAME="E2:24_1_2"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">i</font>,<font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Olive" title="b1">r</font>,<font color="Olive" title="b2">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> where <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</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">i</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">r</font> &amp; <font color="Olive" title="b1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <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> 1,<font color="Maroon" title="c3">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">s</font> &amp; <font color="Olive" title="b2">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan1b.html#E1:24"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan1b.html#E2:24"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="gobrd11.html#T32" title="GOBRD11:th.32">GOBRD11:32</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">r</font>, <font color="Maroon" title="c8">s</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E4:24_1_2"/><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c7">r</font>,<font color="Maroon" title="c8">s</font></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
 <b>and </b><br/><a NAME="E5:24_1_2"/><i><font color="Green" title="E21">A10</font></i>: 
( <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">i</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">r</font> &amp; <font color="Maroon" title="c7">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 <b>and </b><br/><a NAME="E6:24_1_2"/>
( <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> 1,<font color="Maroon" title="c3">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Maroon" title="c8">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</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> 1,<span class="p2">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  )
 <b>by </b><i><a class="txt" href="jordan1b.html#E5:24"><i><font color="Green" title="E16">A5</font></i></a></i>;<br/><a NAME="E7:24_1_2"/><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">r</font>
 <b>by </b><i><a class="txt" href="jordan1b.html#E4:24_1_2"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/><a NAME="E8:24_1_2"/>
( <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> &amp; <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="24_1_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:24_1_2_1"/><i><font color="Green" title="E23">A12</font></i>: 
(  not <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">m</font> or  not <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_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_2_1_1"/>
( <font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">m</font> or <font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan1b.html#E1:24_1_2_1"><i><font color="Green" title="E23">A12</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="24_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:24_1_2_1_1_1"/>
<font color="Maroon" title="c2">i</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c4">m</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:24_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1b.html#E1:24"><i><font color="Green" title="E12">A1</font></i></a>, <a class="txt" href="jordan1b.html#E3:24"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="jordan1b.html#E6:24"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="jordan1b.html#E7:24"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="jordan1b.html#E8:24"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="jordan1b.html#E5:24_1_2"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="jordan1b.html#E7:24_1_2"><i><font color="Green" title="E22">A11</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="24_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:24_1_2_1_1_2"/><i><font color="Green" title="E24">A13</font></i>: 
<font color="Maroon" title="c4">m</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:24_1_2_1_1_2"/>
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</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/><b>hence </b><a NAME="E3:24_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1b.html#E3:24"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="jordan1b.html#E6:24"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="jordan1b.html#E7:24"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="jordan1b.html#E8:24"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="jordan1b.html#E5:24_1_2"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="jordan1b.html#E7:24_1_2"><i><font color="Green" title="E22">A11</font></i></a>, <a class="txt" href="jordan1b.html#E1:24_1_2_1_1_2"><i><font color="Green" title="E24">A13</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><b>end;</b></div>

</div><b>end;</b></div><b>hence </b><a NAME="E9:24_1_2"/>
( <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> or <font color="Maroon" title="c2">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">m</font> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1 )
 <b>by </b><i><a class="txt" href="jordan1b.html#E23"><i><font color="Green" title="E10">Lm2</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
