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<b>let </b><font color="Maroon">E</font> be  <a href="compts_1.html#V6">compact</a> non <a href="sppol_1.html#V1">horizontal</a> non <a href="sppol_1.html#V2">vertical</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">j</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<a NAME="E1:38"/><i><font color="Green">E47</font></i>: 
 <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan8.html#D1">JORDAN8:def 1</a></i>;<br/>
<b>assume </b><a NAME="E2:38"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/>

<a NAME="E3:38"/><b>then </b>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">j</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="jordan8.html#T19">JORDAN8:19</a></i>;<br/>
<a NAME="E4:38"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">E</font> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="subset_1.html#T43">SUBSET_1:43</a></i>;<br/>
<a NAME="E5:38"/><i><font color="Green">E50</font></i>: 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">j</font> is <a href="connsp_1.html#V2">connected</a>
 
<b>by </b><i>, , <a class="ref" href="jordan1a.html#T46">JORDAN1A:46</a></i>;<br/>
<a NAME="E6:38"/>
<font color="Maroon">E</font> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T73">JORDAN2C:73</a></i>;<br/>
<a NAME="E7:38"/><b>then </b><i><font color="Green">E51</font></i>: 
not <font color="Maroon">E</font> <a href="subset_1.html#K3">`</a>  is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T74">JORDAN2C:74</a></i>;<br/>
<a NAME="E8:38"/>
not  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">j</font> is <a href="xboole_0.html#V1">empty</a>
 
<b>by </b><i>, , <a class="ref" href="jordan1a.html#T45">JORDAN1A:45</a></i>;<br/>
<b>then consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E10:38"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">W</font> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">E</font> <a href="subset_1.html#K3">`</a> 
 <b>and </b><br/><a NAME="E11:38"/><i><font color="Green">E53</font></i>: 
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">j</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">W</font>
 <b>by </b><i>, , , <a class="ref" href="goboard9.html#T5">GOBOARD9:5</a></i>;<br/>
<a NAME="E12:38"/>
not  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">j</font> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E13:38"/><b>then </b>
not <font color="Maroon">W</font> is <a href="jordan2c.html#V1">Bounded</a>
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#T16">JORDAN2C:16</a></i>;<br/>
<a NAME="E14:38"/><b>then </b>
<font color="Maroon">W</font> <a href="jordan2c.html#R2">is_outside_component_of</a> <font color="Maroon">E</font>
 
<b>by </b><i>, <a class="ref" href="jordan2c.html#D4">JORDAN2C:def 4</a></i>;<br/>
<a NAME="E15:38"/><b>then </b>
<font color="Maroon">W</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T27">JORDAN2C:27</a></i>;<br/>
<b>hence </b><a NAME="E16:38"/>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">E</font>,<font color="Maroon">n</font></span>)</span></span>)</span>,<font color="Maroon">j</font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">E</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


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