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<b>let </b><font color="Maroon" title="c1">C</font> be  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a> non <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> non <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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>let </b><font color="Maroon" title="c2">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>given </b><font color="Maroon" title="c3">i</font>, <font color="Maroon" title="c4">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><a NAME="E1:47"/><i><font color="Green" title="E29">A1</font></i>: 
<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E2:47"/><i><font color="Green" title="E30">A2</font></i>: 
<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:47"/><i><font color="Green" title="E31">A3</font></i>: 
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c3">i</font>,<font color="Maroon" title="c4">j</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 ;<br/>

<a NAME="E4:47"/><i><font color="Green" title="E32">A4</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3
 
<b>by </b><i><a class="ref" href="jordan1a.html#T49" title="JORDAN1A:th.49">JORDAN1A:49</a></i>;<br/>
<a NAME="E5:47"/><i><font color="Green" title="E33">A5</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> <font color="Maroon" title="c2">n</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3
 
<b>by </b><i><a class="ref" href="jordan8.html#D1" title="JORDAN8:def.1">JORDAN8:def 1</a></i>;<br/>
<a NAME="E6:47"/><i><font color="Green" title="E34">A6</font></i>: 
<font color="Maroon" title="c4">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> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1b.html#E1:47"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="jordan1b.html#E2:47"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="jordan1b.html#E3:47"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="jordan1b.html#T42" target="_self" title="JORDAN1B:th.42">Th42</a></i>;<br/>
<a NAME="E7:47"/><i><font color="Green" title="E35">A7</font></i>: 
<font color="Maroon" title="c3">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> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan1b.html#E1:47"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="jordan1b.html#E2:47"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="jordan1b.html#E3:47"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="jordan1b.html#T43" target="_self" title="JORDAN1B:th.43">Th43</a></i>;<br/>
<a NAME="E8:47"/><i><font color="Green" title="E36">A8</font></i>: 
2 <a href="card_4.html#K1" title="CARD_4:func.1">|^</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="newton.html#T9" title="NEWTON:th.9">NEWTON:9</a></i>;<br/>
<b>assume </b><a NAME="E9:47"/><i><font color="Green" title="E37">A9</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 ;<br/>

<a NAME="E10:47"/><b>then </b><i><font color="Green" title="E38">A10</font></i>: 
 <a href="matrix_1.html#K1" title="MATRIX_1:func.1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3
 
<b>by </b><i><a class="txt" href="jordan1b.html#E4:47"><i><font color="Green" title="E32">A4</font></i></a>, <a class="txt" href="jordan1b.html#E8:47"><i><font color="Green" title="E36">A8</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<a NAME="E11:47"/><i><font color="Green" title="E39">A11</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 3
 
<b>by </b><i><a class="txt" href="jordan1b.html#E5:47"><i><font color="Green" title="E33">A5</font></i></a>, <a class="txt" href="jordan1b.html#E8:47"><i><font color="Green" title="E36">A8</font></i></a>, <a class="txt" href="jordan1b.html#E9:47"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:47_1"/>
( <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or ( <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 ) or <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 </span><b>by </b><i><a class="txt" href="jordan1b.html#E1:47"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="jordan1b.html#E2:47"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="jordan1b.html#E10:47"><i><font color="Green" title="E38">A10</font></i></a>, <a class="txt" href="jordan1b.html#E11:47"><i><font color="Green" title="E39">A11</font></i></a>, <a class="ref" href="nat_1.html#T29" title="NAT_1:th.29">NAT_1:29</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_1"><b>suppose </b></a><a NAME="E1:47_1_1"/>
( <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:47_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1b.html#E1:47"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="jordan1b.html#E2:47"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="jordan1b.html#E3:47"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="jordan1b.html#T40" target="_self" title="JORDAN1B:th.40">Th40</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_2"><b>suppose </b></a><a NAME="E1:47_1_2"/>
( <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 or <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:47_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1b.html#E1:47"><i><font color="Green" title="E29">A1</font></i></a>, <a class="txt" href="jordan1b.html#E2:47"><i><font color="Green" title="E30">A2</font></i></a>, <a class="txt" href="jordan1b.html#E3:47"><i><font color="Green" title="E31">A3</font></i></a>, <a class="ref" href="jordan1b.html#T41" target="_self" title="JORDAN1B:th.41">Th41</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_3"><b>suppose </b></a><a NAME="E1:47_1_3"/>
( <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 &amp; <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 )
 ;<br/><div class="add"><a NAME="E2:47_1_3"/><b>then </b>
 <a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c1">C</font>,0</span>)</span>,2,2 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan1b.html#E3:47"><i><font color="Green" title="E31">A3</font></i></a>, <a class="txt" href="jordan1b.html#E9:47"><i><font color="Green" title="E37">A9</font></i></a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><b>hence </b><a NAME="E3:47_1_3"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan1b.html#T19" target="_self" title="JORDAN1B:th.19">Th19</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="47_1_4"><b>suppose </b></a><a NAME="E1:47_1_4"/>
( <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <font color="Maroon" title="c3">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:47_1_4"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1b.html#E6:47"><i><font color="Green" title="E34">A6</font></i></a>, <a class="txt" href="jordan1b.html#E7:47"><i><font color="Green" title="E35">A7</font></i></a>, <a class="txt" href="jordan1b.html#E10:47"><i><font color="Green" title="E38">A10</font></i></a>, <a class="txt" href="jordan1b.html#E11:47"><i><font color="Green" title="E39">A11</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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