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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">j</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="c3">G</font> be   <a href="goboard1.html#NM1" title="GOBOARD1:NM.1">Go-board</a>;<br/><b>let </b><font color="Maroon" title="c4">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="E1:16"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="goboard1.html#R1" title="GOBOARD1:pred.1">is_sequence_on</a> <font color="Maroon" title="c3">G</font>
 <b>and </b><br/><a NAME="E2:16"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="topreal1.html#V1" title="TOPREAL1:attr.1">special</a>
 <b>and </b><br/><a NAME="E3:16"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c1">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="c3">G</font>
 <b>and </b><br/><a NAME="E4:16"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c2">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="c3">G</font>
 ;<br/>

<b>assume </b><a NAME="E5:16"/>
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c5">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:16"/><i><font color="Green" title="E21">A5</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font></span>)</span>
 <b>and </b><br/><a NAME="E8:16"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E9:16"/>
 <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tarski.html#K3" title="TARSKI:func.3">union</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">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="c4">f</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="topreal1.html#D6" title="TOPREAL1:def.6">TOPREAL1:def 6</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c6">X</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E11:16"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">X</font>
 <b>and </b><br/><a NAME="E12:16"/><i><font color="Green" title="E24">A8</font></i>: 
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">k</font></span>)</span> where <font color="Olive" title="b1">k</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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">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="c4">f</font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan9.html#E8:16"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="tarski.html#D4" title="TARSKI:def.4">TARSKI:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">k</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="E14:16"/><i><font color="Green" title="E25">A9</font></i>: 
<font color="Maroon" title="c6">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c7">k</font>
 <b>and </b><br/><a NAME="E15:16"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font> &amp; <font color="Maroon" title="c7">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="c4">f</font> )
 <b>by </b><i><a class="txt" href="jordan9.html#E12:16"><i><font color="Green" title="E24">A8</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">p</font> = <font color="Maroon" title="c5">x</font> as   <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> <b>by </b><i><a class="txt" href="jordan9.html#E11:16"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="jordan9.html#E14:16"><i><font color="Green" title="E25">A9</font></i></a></i>;<br/>
<a NAME="E17:16"/><i><font color="Green" title="E26">A10</font></i>: 
 <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</font>,<font color="Maroon" title="c2">j</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="goboard5.html#K1" title="GOBOARD5:func.1">v_strip</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</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="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p2">(<span class="default"><a href="goboard5.html#K2" title="GOBOARD5:func.2">h_strip</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">j</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T46" title="TOPS_1:th.46">TOPS_1:46</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1"/>
(  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c7">k</font> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> or  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c7">k</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> )
 </span><b>by </b><i><a class="txt" href="jordan9.html#E2:16"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="sppol_1.html#T41" title="SPPOL_1:th.41">SPPOL_1:41</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1"><b>suppose </b></a><a NAME="E1:16_1_1"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c7">k</font> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">j0</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="E3:16_1_1"/><i><font color="Green" title="E27">A11</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">j0</font> &amp; <font color="Maroon" title="c9">j0</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="c3">G</font> )
 <b>and </b><br/><a NAME="E4:16_1_1"/><i><font color="Green" title="E28">A12</font></i>: 
for <font color="Olive" title="b1">p</font> being  <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>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c7">k</font> holds <br/><font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j0</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="jordan9.html#T14" target="_self" title="JORDAN9:th.14">Th14</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_1_1"/><i><font color="Green" title="E29">A13</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K2" title="GOBOARD5:func.2">h_strip</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c2">j</font></span>)</span>
 ;<br/><a NAME="E2:16_1_1_1"/><i><font color="Green" title="E30">A14</font></i>: 
( <font color="Maroon" title="c9">j0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">j</font> implies <font color="Maroon" title="c9">j0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c2">j</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#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_1_1"/>
( <font color="Maroon" title="c9">j0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font> or <font color="Maroon" title="c9">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> or <font color="Maroon" title="c9">j0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c9">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</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#E2:16_1_1_1"><i><font color="Green" title="E30">A14</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="16_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_1_1_1_1"/><i><font color="Green" title="E31">A15</font></i>: 
<font color="Maroon" title="c9">j0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c2">j</font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E3:16_1_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:16_1_1_1_1_1"/><b>then </b><i><font color="Green" title="E32">A16</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E4:16"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="jordan9.html#E1:16_1_1_1"><i><font color="Green" title="E29">A13</font></i></a>, <a class="ref" href="goboard6.html#T30" title="GOBOARD6:th.30">GOBOARD6:30</a></i>;<br/><a NAME="E4:16_1_1_1_1_1"/>
0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a></i>;<br/><a NAME="E5:16_1_1_1_1_1"/><b>then </b>
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">G</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="E6:16_1_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c2">j</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j0</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E4:16"><i><font color="Green" title="E20">A4</font></i></a>, <a class="txt" href="jordan9.html#E3:16_1_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="txt" href="jordan9.html#E1:16_1_1_1_1_1"><i><font color="Green" title="E31">A15</font></i></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></i>;<br/><b>hence </b><a NAME="E7:16_1_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E11:16"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="jordan9.html#E14:16"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="jordan9.html#E4:16_1_1"><i><font color="Green" title="E28">A12</font></i></a>, <a class="txt" href="jordan9.html#E3:16_1_1_1_1_1"><i><font color="Green" title="E32">A16</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_1_1_1_2"/>
<font color="Maroon" title="c9">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j0</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E3:16_1_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="txt" href="jordan9.html#E1:16_1_1_1"><i><font color="Green" title="E29">A13</font></i></a>, <a class="ref" href="goboard6.html#T30" title="GOBOARD6:th.30">GOBOARD6:30</a></i>;<br/><b>hence </b><a NAME="E3:16_1_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E11:16"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="jordan9.html#E14:16"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="jordan9.html#E4:16_1_1"><i><font color="Green" title="E28">A12</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:16_1_1_1_1_3"/><i><font color="Green" title="E31">A17</font></i>: 
<font color="Maroon" title="c9">j0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c2">j</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="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:16_1_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:16_1_1_1_1_3"/><b>then </b>
<font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">G</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="E4:16_1_1_1_1_3"/><b>then </b><i><font color="Green" title="E32">A18</font></i>: 
<font color="Maroon" title="c8">p</font> <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="c3">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="c2">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="jordan9.html#E1:16_1_1_1"><i><font color="Green" title="E29">A13</font></i></a>, <a class="ref" href="goboard6.html#T31" title="GOBOARD6:th.31">GOBOARD6:31</a></i>;<br/><a NAME="E5:16_1_1_1_1_3"/>
0 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a></i>;<br/><a NAME="E6:16_1_1_1_1_3"/><b>then </b>
( 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">G</font> &amp; <font color="Maroon" title="c2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <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="E7:16_1_1_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">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="c2">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>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j0</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E3:16_1_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="txt" href="jordan9.html#E1:16_1_1_1_1_3"><i><font color="Green" title="E31">A17</font></i></a>, <a class="ref" href="goboard5.html#T5" title="GOBOARD5:th.5">GOBOARD5:5</a></i>;<br/><b>hence </b><a NAME="E8:16_1_1_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E11:16"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="jordan9.html#E14:16"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="jordan9.html#E4:16_1_1"><i><font color="Green" title="E28">A12</font></i></a>, <a class="txt" href="jordan9.html#E4:16_1_1_1_1_3"><i><font color="Green" title="E32">A18</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1_4"><b>suppose </b></a><a NAME="E1:16_1_1_1_1_4"/><i><font color="Green" title="E31">A19</font></i>: 
<font color="Maroon" title="c9">j0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c2">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:16_1_1"><i><font color="Green" title="E27">A11</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:16_1_1_1_1_4"/><b>then </b>
<font color="Maroon" title="c8">p</font> <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="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> 1,<font color="Maroon" title="c9">j0</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E1:16_1_1_1"><i><font color="Green" title="E29">A13</font></i></a>, <a class="txt" href="jordan9.html#E1:16_1_1_1_1_4"><i><font color="Green" title="E31">A19</font></i></a>, <a class="ref" href="goboard6.html#T31" title="GOBOARD6:th.31">GOBOARD6:31</a></i>;<br/><b>hence </b><a NAME="E4:16_1_1_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E11:16"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="jordan9.html#E14:16"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="jordan9.html#E4:16_1_1"><i><font color="Green" title="E28">A12</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:16_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E7:16"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="jordan9.html#E17:16"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2"><b>suppose </b></a><a NAME="E1:16_1_2"/>
 <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c7">k</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c9">i0</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="E3:16_1_2"/><i><font color="Green" title="E27">A20</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">i0</font> &amp; <font color="Maroon" title="c9">i0</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="c3">G</font> )
 <b>and </b><br/><a NAME="E4:16_1_2"/><i><font color="Green" title="E28">A21</font></i>: 
for <font color="Olive" title="b1">p</font> being  <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>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K4" title="TOPREAL1:func.4">LSeg</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c7">k</font> holds <br/><font color="Olive" title="b1">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="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i0</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E1:16"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="jordan9.html#T15" target="_self" title="JORDAN9:th.15">Th15</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1_2_1"/><i><font color="Green" title="E29">A22</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="tops_1.html#K1" title="TOPS_1:func.1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K1" title="GOBOARD5:func.1">v_strip</a> <font color="Maroon" title="c3">G</font>,<font color="Maroon" title="c1">i</font></span>)</span>
 ;<br/><a NAME="E2:16_1_2_1"/><i><font color="Green" title="E30">A23</font></i>: 
( <font color="Maroon" title="c9">i0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">i</font> implies <font color="Maroon" title="c9">i0</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c1">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#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_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="E1:16_1_2_1_1"/>
( <font color="Maroon" title="c9">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i</font> or <font color="Maroon" title="c9">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> or <font color="Maroon" title="c9">i0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 or <font color="Maroon" title="c9">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</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#E2:16_1_2_1"><i><font color="Green" title="E30">A23</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="16_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:16_1_2_1_1_1"/><i><font color="Green" title="E31">A24</font></i>: 
<font color="Maroon" title="c9">i0</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c1">i</font>
 ;<br/><div class="add"><a NAME="E2:16_1_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1
 <b>by </b><i><a class="txt" href="jordan9.html#E3:16_1_2"><i><font color="Green" title="E27">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:16_1_2_1_1_1"/><b>then </b><i><font color="Green" title="E32">A25</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E3:16"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="jordan9.html#E1:16_1_2_1"><i><font color="Green" title="E29">A22</font></i></a>, <a class="ref" href="goboard6.html#T32" title="GOBOARD6:th.32">GOBOARD6:32</a></i>;<br/><a NAME="E4:16_1_2_1_1_1"/>
0 <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="c3">G</font>
 <b>by </b><i><a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a></i>;<br/><a NAME="E5:16_1_2_1_1_1"/><b>then </b>
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="c3">G</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="E6:16_1_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c1">i</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i0</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E3:16"><i><font color="Green" title="E19">A3</font></i></a>, <a class="txt" href="jordan9.html#E3:16_1_2"><i><font color="Green" title="E27">A20</font></i></a>, <a class="txt" href="jordan9.html#E1:16_1_2_1_1_1"><i><font color="Green" title="E31">A24</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>;<br/><b>hence </b><a NAME="E7:16_1_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E11:16"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="jordan9.html#E14:16"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="jordan9.html#E4:16_1_2"><i><font color="Green" title="E28">A21</font></i></a>, <a class="txt" href="jordan9.html#E3:16_1_2_1_1_1"><i><font color="Green" title="E32">A25</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:16_1_2_1_1_2"/>
<font color="Maroon" title="c9">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font>
 ;<br/><div class="add"><a NAME="E2:16_1_2_1_1_2"/><b>then </b>
<font color="Maroon" title="c8">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i0</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E3:16_1_2"><i><font color="Green" title="E27">A20</font></i></a>, <a class="txt" href="jordan9.html#E1:16_1_2_1"><i><font color="Green" title="E29">A22</font></i></a>, <a class="ref" href="goboard6.html#T32" title="GOBOARD6:th.32">GOBOARD6:32</a></i>;<br/><b>hence </b><a NAME="E3:16_1_2_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E11:16"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="jordan9.html#E14:16"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="jordan9.html#E4:16_1_2"><i><font color="Green" title="E28">A21</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1_1_3"><b>suppose </b></a><a NAME="E1:16_1_2_1_1_3"/><i><font color="Green" title="E31">A26</font></i>: 
<font color="Maroon" title="c9">i0</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:16_1_2_1_1_3"/><b>then </b>
<font color="Maroon" title="c1">i</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">G</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:16_1_2"><i><font color="Green" title="E27">A20</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E3:16_1_2_1_1_3"/><b>then </b>
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</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="E4:16_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E32">A27</font></i>: 
<font color="Maroon" title="c8">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">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>by </b><i><a class="txt" href="jordan9.html#E1:16_1_2_1"><i><font color="Green" title="E29">A22</font></i></a>, <a class="ref" href="goboard6.html#T33" title="GOBOARD6:th.33">GOBOARD6:33</a></i>;<br/><a NAME="E5:16_1_2_1_1_3"/>
0 <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="c3">G</font>
 <b>by </b><i><a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a></i>;<br/><a NAME="E6:16_1_2_1_1_3"/><b>then </b>
( 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="c3">G</font> &amp; <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 )
 <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="E7:16_1_2_1_1_3"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">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>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i0</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E3:16_1_2"><i><font color="Green" title="E27">A20</font></i></a>, <a class="txt" href="jordan9.html#E1:16_1_2_1_1_3"><i><font color="Green" title="E31">A26</font></i></a>, <a class="ref" href="goboard5.html#T4" title="GOBOARD5:th.4">GOBOARD5:4</a></i>;<br/><b>hence </b><a NAME="E8:16_1_2_1_1_3"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E11:16"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="jordan9.html#E14:16"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="jordan9.html#E4:16_1_2"><i><font color="Green" title="E28">A21</font></i></a>, <a class="txt" href="jordan9.html#E4:16_1_2_1_1_3"><i><font color="Green" title="E32">A27</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_2_1_1_4"><b>suppose </b></a><a NAME="E1:16_1_2_1_1_4"/><i><font color="Green" title="E31">A28</font></i>: 
<font color="Maroon" title="c9">i0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 ;<br/><div class="add"><a NAME="E2:16_1_2_1_1_4"/><b>then </b>
<font color="Maroon" title="c1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="txt" href="jordan9.html#E3:16_1_2"><i><font color="Green" title="E27">A20</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E3:16_1_2_1_1_4"/><b>then </b>
<font color="Maroon" title="c8">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">G</font> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c9">i0</font>,1</span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 <b>by </b><i><a class="txt" href="jordan9.html#E1:16_1_2_1"><i><font color="Green" title="E29">A22</font></i></a>, <a class="txt" href="jordan9.html#E1:16_1_2_1_1_4"><i><font color="Green" title="E31">A28</font></i></a>, <a class="ref" href="goboard6.html#T33" title="GOBOARD6:th.33">GOBOARD6:33</a></i>;<br/><b>hence </b><a NAME="E4:16_1_2_1_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E11:16"><i><font color="Green" title="E23">A7</font></i></a>, <a class="txt" href="jordan9.html#E14:16"><i><font color="Green" title="E25">A9</font></i></a>, <a class="txt" href="jordan9.html#E4:16_1_2"><i><font color="Green" title="E28">A21</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E6:16_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan9.html#E7:16"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="jordan9.html#E17:16"><i><font color="Green" title="E26">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
