<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:37"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">i</font> <a href="jordan1h.html#R1">is_sufficiently_large_for</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:37"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E3:37"/><i><font color="Green">E43</font></i>: 
not  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2">LeftComp</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E4:37"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">j</font> <a href="jordan1h.html#R1">is_sufficiently_large_for</a> <font color="Maroon">C</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E5:37"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font>
 
<b>by </b><i><a class="ref" href="jordan13.html#D1">JORDAN13:def 1</a></i>;<br/>
<a NAME="E6:37"/><i><font color="Green">E53</font></i>: 
 <a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font> <a href="goboard1.html#R1">is_sequence_on</a>  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font>
 
<b>by </b><i>, <a class="ref" href="jordan13.html#D1">JORDAN13:def 1</a></i>;<br/>
<b>consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</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="E8:37"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>
 <b>and </b><br/><a NAME="E9:37"/><i><font color="Green">E55</font></i>: 
not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2">LeftComp</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="subset_1.html#T7">SUBSET_1:7</a></i>;<br/>
<b>reconsider </b><font color="Maroon">D</font> = <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a>  as   <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>consider </b><font color="Maroon">i1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:37"/><i><font color="Green">E56</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font>
 <b>and </b><br/><a NAME="E13:37"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>
 <b>and </b><br/><a NAME="E14:37"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>
 <b>by </b><i>, <a class="ref" href="sppol_2.html#T13">SPPOL_2:13</a></i>;<br/>
<a NAME="E15:37"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">i1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font>;<br/>
<b>set </b><font color="Maroon">G</font> =  <a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font>;<br/>
<div><i><font color="Green">E67</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:37_1"/><i><font color="Green">E68</font></i>: 
not  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:37_1"/>
 ex <font color="Olive">i2</font>, <font color="Olive">j2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i2</font> &amp; <font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1 <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">C</font>,<font color="Maroon">i</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j2</font> &amp; <font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>,<font color="Olive">i2</font>,<font color="Olive">j2</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:37_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">i1</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan14.html#E2:37"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">i4</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j4</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:37_1_1"/><i><font color="Green">E70</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i4</font>,<font color="Maroon">j4</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>
 <b>and </b><br/><a NAME="E4:37_1_1"/><i><font color="Green">E71</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i4</font>,<font color="Maroon">j4</font>
 <b>by </b><i>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/>
<a NAME="E5:37_1_1"/><i><font color="Green">E72</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i4</font> &amp; <font color="Maroon">i4</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">C</font>,<font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan14.html#E5:37"><i><font color="Green">E45</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E6:37_1_1"/><i><font color="Green">E73</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j4</font> &amp; <font color="Maroon">j4</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan14.html#E5:37"><i><font color="Green">E45</font></i></a>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E7:37_1_1"/><i><font color="Green">E74</font></i>: 
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E8:37_1_1"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/>
<b>then consider </b><font color="Maroon">i5</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j5</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:37_1_1"/><i><font color="Green">E76</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">i5</font>,<font color="Maroon">j5</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a>  <a href="matrix_1.html#K2">Indices</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>
 <b>and </b><br/><a NAME="E11:37_1_1"/><i><font color="Green">E77</font></i>: 
<span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <span class="p1">(<span class="default"><font color="Maroon">i1</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">i5</font>,<font color="Maroon">j5</font>
 <b>by </b><i>, <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/>
<a NAME="E12:37_1_1"/><i><font color="Green">E78</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i5</font> &amp; <font color="Maroon">i5</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">C</font>,<font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E13:37_1_1"/><i><font color="Green">E79</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j5</font> &amp; <font color="Maroon">j5</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="goboard5.html#T1">GOBOARD5:1</a></i>;<br/>
<a NAME="E14:37_1_1"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>
 
;<br/>
<a NAME="E15:37_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
( ( <font color="Maroon">i4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i5</font> &amp; <font color="Maroon">j4</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j5</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">i4</font> <a href="binarith.html#K3">-'</a> 1</span>)</span>,<font color="Maroon">j4</font> ) or ( <font color="Maroon">i4</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i5</font> &amp; <font color="Maroon">j4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j5</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i4</font>,<font color="Maroon">j4</font> ) or ( <font color="Maroon">i4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i5</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j5</font> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i5</font>,<span class="p1">(<span class="default"><font color="Maroon">j5</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> ) or ( <font color="Maroon">i4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i5</font> &amp; <font color="Maroon">j4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j5</font> <a href="nat_1.html#K1">+</a> 1 &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i4</font>,<font color="Maroon">j5</font> ) )
 
<b>by </b><i>, <a class="txt" href="jordan14.html#E2:37"><i><font color="Green">E38</font></i></a>, , <a class="txt" href="jordan14.html#E5:37"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jordan14.html#T4" target="_self">Th4</a>, , , <a class="ref" href="gobrd13.html#D3">GOBRD13:def 3</a></i>;<br/>
<a NAME="E16:37_1_1"/>
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">i4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i5</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">j4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j5</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i>, <a class="txt" href="jordan14.html#E4:37"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan14.html#E5:37"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jordan14.html#T4" target="_self">Th4</a>, , , , <a class="ref" href="goboard1.html#D11">GOBOARD1:def 11</a></i>;<br/>
<a NAME="E17:37_1_1"/><b>then </b><i><font color="Green">E81</font></i>: 
( (  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">i4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">i5</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">j4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j5</font> ) or (  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">j4</font> <a href="real_1.html#K5">-</a> <font color="Maroon">j5</font></span>)</span> <a href="hidden.html#R1">=</a> 1 &amp; <font color="Maroon">i4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i5</font> ) )
 
<b>by </b><i><a class="ref" href="goboard1.html#T2">GOBOARD1:2</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1_1_1"/>
( ( <font color="Maroon">i4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i5</font> &amp; <font color="Maroon">j4</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j5</font> ) or ( <font color="Maroon">i4</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i5</font> &amp; <font color="Maroon">j4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j5</font> ) or ( <font color="Maroon">i4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i5</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j5</font> ) or ( <font color="Maroon">i4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i5</font> &amp; <font color="Maroon">j4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j5</font> <a href="nat_1.html#K1">+</a> 1 ) )
 </span><b>by </b><i><a class="txt" href="jordan14.html#E12:37"><i><font color="Green">E56</font></i></a>, <a class="ref" href="goboard1.html#T1">GOBOARD1:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_1"><b>suppose </b></a><a NAME="E1:37_1_1_1_1"/><i><font color="Green">E82</font></i>: 
( <font color="Maroon">i4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i5</font> &amp; <font color="Maroon">j4</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j5</font> )
 ;<br/><div class="add"><a NAME="E2:37_1_1_1_1"/><i><font color="Green">E83</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">i4</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i4</font>
 <b>by </b><i>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E3:37_1_1_1_1"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i4</font>
 <b>by </b><i><a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan14.html#E2:37"><i><font color="Green">E38</font></i></a>, , <a class="txt" href="jordan14.html#E5:37"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jordan14.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E4:37_1_1_1_1"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i4</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:37_1_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i4</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan5b.html#T2">JORDAN5B:2</a></i>;<br/><b>hence </b><a NAME="E6:37_1_1_1_1"/>
 ex <font color="Olive">i2</font>, <font color="Olive">j2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i2</font> &amp; <font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1 <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">C</font>,<font color="Maroon">i</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j2</font> &amp; <font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>,<font color="Olive">i2</font>,<font color="Olive">j2</font> )
 <b>by </b><i>, , , <a class="txt" href="jordan14.html#E8:37"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan14.html#E9:37"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan14.html#E13:37"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan14.html#E14:37"><i><font color="Green">E58</font></i></a>, <a class="ref" href="jordan1h.html#T44">JORDAN1H:44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_2"><b>suppose </b></a><a NAME="E1:37_1_1_1_2"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">i4</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">i5</font> &amp; <font color="Maroon">j4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j5</font> )
 ;<br/><div class="add"><a NAME="E2:37_1_1_1_2"/>
<font color="Maroon">j4</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan14.html#E2:37"><i><font color="Green">E38</font></i></a>, , <a class="txt" href="jordan14.html#E5:37"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jordan14.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E3:37_1_1_1_2"/><b>then </b>
<font color="Maroon">j4</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:37_1_1_1_2"/>
 ex <font color="Olive">i2</font>, <font color="Olive">j2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i2</font> &amp; <font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1 <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">C</font>,<font color="Maroon">i</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j2</font> &amp; <font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>,<font color="Olive">i2</font>,<font color="Olive">j2</font> )
 <b>by </b><i>, , , <a class="txt" href="jordan14.html#E6:37"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan14.html#E9:37"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan14.html#E13:37"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan1h.html#T44">JORDAN1H:44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_3"><b>suppose </b></a><a NAME="E1:37_1_1_1_3"/><i><font color="Green">E85</font></i>: 
( <font color="Maroon">i4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i5</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">j4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j5</font> )
 ;<br/><div class="add"><a NAME="E2:37_1_1_1_3"/><i><font color="Green">E86</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">j5</font> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">j5</font>
 <b>by </b><i><a class="txt" href="jordan14.html#E8:37"><i><font color="Green">E54</font></i></a>, <a class="ref" href="binarith.html#T53">BINARITH:53</a></i>;<br/><a NAME="E3:37_1_1_1_3"/>
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j5</font>
 <b>by </b><i><a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, , , , , </i>;<br/><a NAME="E4:37_1_1_1_3"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j5</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:37_1_1_1_3"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j5</font> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan5b.html#T2">JORDAN5B:2</a></i>;<br/><b>hence </b><a NAME="E6:37_1_1_1_3"/>
 ex <font color="Olive">i2</font>, <font color="Olive">j2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i2</font> &amp; <font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1 <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">C</font>,<font color="Maroon">i</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j2</font> &amp; <font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>,<font color="Olive">i2</font>,<font color="Olive">j2</font> )
 <b>by </b><i>, , <a class="txt" href="jordan14.html#E6:37"><i><font color="Green">E53</font></i></a>, <a class="txt" href="jordan14.html#E8:37"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan14.html#E9:37"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan14.html#E13:37"><i><font color="Green">E57</font></i></a>, <a class="txt" href="jordan14.html#E14:37"><i><font color="Green">E58</font></i></a>, <a class="ref" href="jordan1h.html#T44">JORDAN1H:44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1_1_4"><b>suppose </b></a><a NAME="E1:37_1_1_1_4"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">i4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i5</font> &amp; <font color="Maroon">j4</font> <a href="hidden.html#R1">=</a> <font color="Maroon">j5</font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/><div class="add"><a NAME="E2:37_1_1_1_4"/>
<font color="Maroon">i4</font> <a href="xxreal_0.html#NR3" target="_self">&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">C</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan14.html#E2:37"><i><font color="Green">E38</font></i></a>, , <a class="txt" href="jordan14.html#E5:37"><i><font color="Green">E45</font></i></a>, <a class="ref" href="jordan14.html#T4" target="_self">Th4</a>, </i>;<br/><a NAME="E3:37_1_1_1_4"/><b>then </b>
<font color="Maroon">i4</font> <a href="nat_1.html#K1">+</a> 1 <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">C</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E4:37_1_1_1_4"/>
 ex <font color="Olive">i2</font>, <font color="Olive">j2</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">i2</font> &amp; <font color="Olive">i2</font> <a href="nat_1.html#K1">+</a> 1 <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">C</font>,<font color="Maroon">i</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j2</font> &amp; <font color="Olive">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span> &amp;  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>,<font color="Olive">i2</font>,<font color="Olive">j2</font> )
 <b>by </b><i>, , , <a class="txt" href="jordan14.html#E8:37"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan14.html#E9:37"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan14.html#E13:37"><i><font color="Green">E57</font></i></a>, <a class="ref" href="jordan1h.html#T44">JORDAN1H:44</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>then consider </b><font color="Maroon">i2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:37_1"/><i><font color="Green">E88</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i2</font> &amp; <font color="Maroon">i2</font> <a href="nat_1.html#K1">+</a> 1 <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">C</font>,<font color="Maroon">i</font></span>)</span> )
 <b>and </b><br/><a NAME="E5:37_1"/><i><font color="Green">E89</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j2</font> &amp; <font color="Maroon">j2</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span> )
 <b>and </b><br/><a NAME="E6:37_1"/><i><font color="Green">E90</font></i>: 
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font>
 ;<br/><a NAME="E7:37_1"/><i><font color="Green">E91</font></i>: 
<font color="Maroon">i2</font> <a href="xxreal_0.html#NR3" target="_self">&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">C</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E4:37"><i><font color="Green">E44</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E8:37_1"/><i><font color="Green">E92</font></i>: 
<font color="Maroon">j2</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="matrix_1.html#K1">width</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E5:37"><i><font color="Green">E45</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E9:37_1"/><i><font color="Green">E93</font></i>: 
not  <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E3:37"><i><font color="Green">E43</font></i></a>, <a class="ref" href="jordan14.html#T4" target="_self">Th4</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p3">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2">LeftComp</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p3">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p2">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p3">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2">LeftComp</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="gobrd14.html#T31">GOBRD14:31</a></i>
<br/>.= 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>
<b>by </b><i><a class="ref" href="gobrd14.html#T25">GOBRD14:25</a></i>
;<br/><a NAME="E11:37_1"/><b>then </b><i><font color="Green">E94</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">C</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T73">XBOOLE_1:73</a></i>;<br/><a NAME="E12:37_1"/>
 <a href="goboard5.html#K3">cell</a> <span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="gobrd11.html#T35">GOBRD11:35</a></i>;<br/><a NAME="E13:37_1"/><b>then </b><i><font color="Green">E95</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="tsep_1.html#T40">TSEP_1:40</a></i>;<br/><a NAME="E14:37_1"/><i><font color="Green">E96</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i>, , , </i>;<br/><a NAME="E15:37_1"/><i><font color="Green">E97</font></i>: 
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span> <a href="connsp_1.html#R4">is_a_component_of</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="goboard9.html#D1">GOBOARD9:def 1</a></i>;<br/><a NAME="E16:37_1"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i>, , <a class="ref" href="goboard9.html#T21">GOBOARD9:21</a></i>;<br/><a NAME="E17:37_1"/><b>then </b><i><font color="Green">E98</font></i>: 
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i>, , <a class="txt" href="jordan14.html#E6:37"><i><font color="Green">E53</font></i></a>, <a class="ref" href="goboard9.html#T6">GOBOARD9:6</a></i>;<br/><a NAME="E18:37_1"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="goboard5.html#K3">cell</a> <span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>,<font color="Maroon">i2</font>,<font color="Maroon">j2</font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan14.html#T4" target="_self">Th4</a>, <a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/><a NAME="E19:37_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p2">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E8:37"><i><font color="Green">E54</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E20:37_1"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="gobrd13.html#K4">left_cell</a> <span class="p3">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p3">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2">LeftComp</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/><a NAME="E21:37_1"/><b>then </b><i><font color="Green">E99</font></i>: 
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2">LeftComp</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan14.html#E2:37"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="jordan9.html#T13">JORDAN9:13</a></i>;<br/><a NAME="E22:37_1"/>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font> <a href="tarski.html#R1">c=</a>  <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>
 <b>by </b><i>, <a class="txt" href="jordan14.html#E2:37"><i><font color="Green">E38</font></i></a>, , <a class="ref" href="jordan1h.html#T26">JORDAN1H:26</a></i>;<br/><a NAME="E23:37_1"/><b>then </b>
 <a href="topreal1.html#K4">LSeg</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2">LeftComp</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jordan14.html#E9:37"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>hence </b><a NAME="E24:37_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan14.html#E1:37"><i><font color="Green">E37</font></i></a>, </i>;<br/></div><b>end;</b></div>
<a NAME="E17:37"/><i><font color="Green">E100</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E18:37"/>
 <a href="gobrd13.html#K4">left_cell</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span>,<font color="Maroon">i1</font>,<span class="p1">(<span class="default"><a href="jordan8.html#K1">Gauge</a> <font color="Maroon">C</font>,<font color="Maroon">j</font></span>)</span> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="jordan14.html#T1" target="_self">Th1</a>, <a class="txt" href="jordan14.html#E2:37"><i><font color="Green">E38</font></i></a>, , </i>;<br/>
<a NAME="E19:37"/><b>then </b><i><font color="Green">E101</font></i>: 
<font color="Maroon">C</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E20:37"/><i><font color="Green">E102</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="topreal1.html#K5">L~</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="gobrd14.html#T31">GOBRD14:31</a></i>;<br/>
<a NAME="E21:37"/>
<font color="Maroon">C</font> <a href="subset_1.html#R1">misses</a>  <a href="topreal1.html#K5">L~</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E22:37"/><b>then </b><i><font color="Green">E103</font></i>: 
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan14.html#E4:37"><i><font color="Green">E44</font></i></a>, <a class="txt" href="jordan14.html#E5:37"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_1.html#T70">XBOOLE_1:70</a></i>;<br/>
<a NAME="E23:37"/><i><font color="Green">E104</font></i>: 
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="goboard9.html#D2">GOBOARD9:def 2</a></i>;<br/>
<a NAME="E24:37"/>
<font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="goboard9.html#K3">RightComp</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="goboard9.html#K2">LeftComp</a> <span class="p2">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="gobrd12.html#T11">GOBRD12:11</a></i>;<br/>
<a NAME="E25:37"/><b>then </b>
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E26:37"/><b>then </b><i><font color="Green">E105</font></i>: 
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="txt" href="jordan14.html#E3:37"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E27:37"/><i><font color="Green">E106</font></i>: 
 <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span> <a href="connsp_1.html#R4">is_a_component_of</a> <font color="Maroon">D</font>
 
<b>by </b><i><a class="ref" href="goboard9.html#D1">GOBOARD9:def 1</a></i>;<br/>
<a NAME="E28:37"/>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a> <font color="Maroon">C</font>
 
;<br/>
<a NAME="E29:37"/><b>then </b>
<font color="Maroon">C</font> <a href="subset_1.html#R2">meets</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan14.html#E3:37"><i><font color="Green">E43</font></i></a>, <a class="ref" href="xboole_1.html#T63">XBOOLE_1:63</a></i>;<br/>
<a NAME="E30:37"/><b>then </b>
 <a href="goboard9.html#K3">RightComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="goboard9.html#K2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1">Span</a> <font color="Maroon">C</font>,<font color="Maroon">i</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan14.html#T4" target="_self">Th4</a>, , , , <a class="ref" href="jordan9.html#T3">JORDAN9:3</a></i>;<br/>
<b>hence </b><a NAME="E31:37"/>
contradiction
 <b>by </b><i><a class="ref" href="sprect_4.html#T7">SPRECT_4:7</a></i>;<br/>


</div>
