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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">i</font>, <font color="Maroon" title="c3">n</font>, <font color="Maroon" title="c4">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>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green" title="E26">A1</font></i>: 
<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> <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="c3">n</font></span>)</span>
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green" title="E27">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="c3">n</font></span>)</span>
 <b>and </b><br/><a NAME="E3:42"/><i><font color="Green" title="E28">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="c3">n</font></span>)</span>,<font color="Maroon" title="c2">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>
 <b>and </b><br/><a NAME="E4:42"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1"/>
( <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><b>by </b><i><a class="txt" href="jordan1b.html#E4:42"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><b>suppose </b></a><a NAME="E1:42_1_1"/>
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1b.html#E1:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="jordan1b.html#E3:42"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="jordan1b.html#E43"><i><font color="Green" title="E22">Lm4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2"><b>suppose </b></a><a NAME="E1:42_1_2"/><i><font color="Green" title="E30">A5</font></i>: 
<font color="Maroon" title="c4">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:42_1_2"/><i><font color="Green" title="E31">A6</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="c3">n</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="goboard1.html#D5" title="GOBOARD1:def.5">GOBOARD1:def 5</a></i>;<br/><a NAME="E3:42_1_2"/><b>then </b><i><font color="Green" title="E32">A7</font></i>: 
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</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="c3">n</font></span>)</span>
 <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="E4:42_1_2"/><b>then </b><i><font color="Green" title="E33">A8</font></i>: 
not  <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="c3">n</font></span>)</span>,<font color="Maroon" title="c2">i</font>,1 is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 <b>by </b><i><a class="txt" href="jordan1b.html#E1:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="jordan1a.html#T45" title="JORDAN1A:th.45">JORDAN1A:45</a></i>;<br/><a NAME="E5:42_1_2"/><i><font color="Green" title="E34">A9</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="c3">n</font></span>)</span>,<font color="Maroon" title="c2">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan1b.html#E1:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="jordan1a.html#T70" title="JORDAN1A:th.70">JORDAN1A:70</a></i>;<br/><a NAME="E6:42_1_2"/><i><font color="Green" title="E35">A10</font></i>: 
<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> <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="c3">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1b.html#E1:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="jordan1b.html#E2:42"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="jordan1b.html#E3:42"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="jordan1b.html#E44"><i><font color="Green" title="E23">Lm5</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E7:42_1_2"/><i><font color="Green" title="E36">A11</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <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="c3">n</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1b.html#E2:42_1_2"><i><font color="Green" title="E31">A6</font></i></a></i>;<br/><a NAME="E8:42_1_2"/>
<font color="Maroon" title="c2">i</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="txt" href="jordan1b.html#E2:42"><i><font color="Green" title="E27">A2</font></i></a>, <a class="txt" href="jordan1b.html#E3:42"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="jordan1b.html#E42"><i><font color="Green" title="E21">Lm3</font></i></a></i>;<br/><a NAME="E9:42_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<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="c3">n</font></span>)</span>,<font color="Maroon" title="c2">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3" title="GOBOARD5:func.3">cell</a> <span class="p2">(<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="c3">n</font></span>)</span>,<font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <span class="p1">(<span class="default"><span class="p2">(<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="c3">n</font></span>)</span> <a href="matrix_1.html#K3" title="MATRIX_1:func.3">*</a> <font color="Maroon" title="c2">i</font>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<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="c3">n</font></span>)</span> <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#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan1b.html#E6:42_1_2"><i><font color="Green" title="E35">A10</font></i></a>, <a class="txt" href="jordan1b.html#E7:42_1_2"><i><font color="Green" title="E36">A11</font></i></a>, <a class="ref" href="goboard5.html#T27" title="GOBOARD5:th.27">GOBOARD5:27</a>, <a class="ref" href="nat_1.html#T14" title="NAT_1:th.14">NAT_1:14</a></i>;<br/><a NAME="E10:42_1_2"/><b>then </b><i><font color="Green" title="E37">A12</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="c3">n</font></span>)</span>,<font color="Maroon" title="c2">i</font>,<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <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="c3">n</font></span>)</span>,<font color="Maroon" title="c2">i</font>,<span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/><a NAME="E11:42_1_2"/>
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="jordan2c.html#T73" title="JORDAN2C:th.73">JORDAN2C:73</a>, <a class="ref" href="jordan2c.html#T76" title="JORDAN2C:th.76">JORDAN2C:76</a></i>;<br/><a NAME="E12:42_1_2"/><b>then </b><i><font color="Green" title="E38">A13</font></i>: 
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#D4" title="JORDAN2C:def.4">JORDAN2C:def 4</a></i>;<br/><a NAME="E13:42_1_2"/>
 <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="ref" href="jordan2c.html#T29" title="JORDAN2C:th.29">JORDAN2C:29</a></i>;<br/><a NAME="E14:42_1_2"/><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>,<font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c2">i</font>,1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>by </b><i><a class="txt" href="jordan1b.html#E3:42"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="jordan1b.html#E1:42_1_2"><i><font color="Green" title="E30">A5</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E15:42_1_2"/><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>,<font color="Maroon" title="c3">n</font></span>)</span>,<font color="Maroon" title="c2">i</font>,1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="txt" href="jordan1b.html#E1:42"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="jordan1b.html#E3:42_1_2"><i><font color="Green" title="E32">A7</font></i></a>, <a class="txt" href="jordan1b.html#E5:42_1_2"><i><font color="Green" title="E34">A9</font></i></a>, <a class="txt" href="jordan1b.html#E10:42_1_2"><i><font color="Green" title="E37">A12</font></i></a>, <a class="txt" href="jordan1b.html#E12:42_1_2"><i><font color="Green" title="E38">A13</font></i></a>, <a class="ref" href="goboard9.html#T6" title="GOBOARD9:th.6">GOBOARD9:6</a>, <a class="ref" href="jordan1a.html#T46" title="JORDAN1A:th.46">JORDAN1A:46</a></i>;<br/><b>hence </b><a NAME="E16:42_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan1b.html#E3:42"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="jordan1b.html#E1:42_1_2"><i><font color="Green" title="E30">A5</font></i></a>, <a class="txt" href="jordan1b.html#E4:42_1_2"><i><font color="Green" title="E33">A8</font></i></a>, <a class="ref" href="jordan2c.html#T28" title="JORDAN2C:th.28">JORDAN2C:28</a>, <a class="ref" href="xboole_1.html#T68" title="XBOOLE_1:th.68">XBOOLE_1:68</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
