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<b>let </b><font color="Maroon" title="c1">i</font>, <font color="Maroon" title="c2">n</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="c3">C</font> be  <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>assume </b><a NAME="E1:63"/><i><font color="Green" title="E38">A1</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> <span class="p1">(<span class="default"><a href="jordan8.html#K1" title="JORDAN8:func.1">Gauge</a> <font color="Maroon" title="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 ;<br/>

<a NAME="E2:63"/><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="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">i</font>,0 <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="ref" href="jordan8.html#T20" title="JORDAN8:th.20">JORDAN8:20</a></i>;<br/>
<a NAME="E3:63"/><b>then </b><i><font color="Green" title="E39">A2</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="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">i</font>,0 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T43" title="SUBSET_1:th.43">SUBSET_1:43</a></i>;<br/>
<a NAME="E4:63"/><i><font color="Green" title="E40">A3</font></i>: 
0 <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="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>
 
;<br/>
<a NAME="E5:63"/><b>then </b><i><font color="Green" title="E41">A4</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="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">i</font>,0 is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E1:63"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="jordan1a.html#T46" target="_self" title="JORDAN1A:th.46">Th46</a></i>;<br/>
<a NAME="E6:63"/><i><font color="Green" title="E42">A5</font></i>: 
not <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T74" title="JORDAN2C:th.74">JORDAN2C:74</a></i>;<br/>
<a NAME="E7:63"/>
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="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">i</font>,0 is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E1:63"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="jordan1a.html#E4:63"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="jordan1a.html#T45" target="_self" title="JORDAN1A:th.45">Th45</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">W</font> being   <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><b> such that </b><br/><a NAME="E9:63"/><i><font color="Green" title="E43">A6</font></i>: 
<font color="Maroon" title="c4">W</font> <a href="connsp_1.html#R4" title="CONNSP_1:pred.4">is_a_component_of</a> <font color="Maroon" title="c3">C</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 <b>and </b><br/><a NAME="E10:63"/><i><font color="Green" title="E44">A7</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="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">i</font>,0 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">W</font>
 <b>by </b><i><a class="txt" href="jordan1a.html#E3:63"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="jordan1a.html#E5:63"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="jordan1a.html#E6:63"><i><font color="Green" title="E42">A5</font></i></a>, <a class="ref" href="goboard9.html#T5" title="GOBOARD9:th.5">GOBOARD9:5</a></i>;<br/>
<a NAME="E11:63"/>
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="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">i</font>,0 is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E1:63"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="jordan1a.html#T47" target="_self" title="JORDAN1A:th.47">Th47</a></i>;<br/>
<a NAME="E12:63"/><b>then </b>
not <font color="Maroon" title="c4">W</font> is <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E10:63"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="jordan2c.html#T16" title="JORDAN2C:th.16">JORDAN2C:16</a></i>;<br/>
<a NAME="E13:63"/><b>then </b>
<font color="Maroon" title="c4">W</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Maroon" title="c3">C</font>
 
<b>by </b><i><a class="txt" href="jordan1a.html#E9:63"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="jordan2c.html#D4" title="JORDAN2C:def.4">JORDAN2C:def 4</a></i>;<br/>
<a NAME="E14:63"/><b>then </b>
<font color="Maroon" title="c4">W</font> <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="c3">C</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T27" title="JORDAN2C:th.27">JORDAN2C:27</a></i>;<br/>
<b>hence </b><a NAME="E15:63"/>
 <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="c3">C</font>,<font color="Maroon" title="c2">n</font></span>)</span>,<font color="Maroon" title="c1">i</font>,0 <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="c3">C</font>
 <b>by </b><i><a class="txt" href="jordan1a.html#E10:63"><i><font color="Green" title="E44">A7</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


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