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<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">C</font> be  <a href="connsp_1.html#V2">connected</a> <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>consider </b><font color="Maroon">k</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">t</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="E2:73"/><i><font color="Green">E64</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> &amp; <font color="Maroon">k</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="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</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">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">k</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">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">t</font>,1 )
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1"/>
( <font color="Maroon">k</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="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> or <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 </span><b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_1"><b>suppose </b></a><a NAME="E1:73_1_1"/>
<font color="Maroon">k</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="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:73_1_1"/>
 ex <font color="Olive">k</font>, <font color="Olive">t</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">k</font> &amp; <font color="Olive">k</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="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">t</font> &amp; <font color="Olive">t</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">n</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Olive">k</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">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Olive">t</font>,1 )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_2"><b>suppose </b></a><a NAME="E1:73_1_2"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>take </b>
<font color="Maroon">t</font>
;<br/><a NAME="E2:73_1_2"/>
4 <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span>
 <b>by </b><i><a class="ref" href="goboard7.html#T36">GOBOARD7:36</a></i>;<br/><b>hence </b><a NAME="E3:73_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 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="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>thus </b><a NAME="E4:73_1_2"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">t</font> &amp; <font color="Maroon">t</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">n</font></span>)</span> )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E5:73_1_2"/>
<span class="p1">(<span class="default"><a href="jordan9.html#K1">Cage</a> <font color="Maroon">C</font>,<font color="Maroon">n</font></span>)</span> <a href="finseq_4.html#K4">/.</a> 1 <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">n</font></span>)</span> <a href="matrix_1.html#K3">*</a> <font color="Maroon">t</font>,1
 <b>by </b><i>, <a class="txt" href="jordan1a.html#E2"><i><font color="Green">Lemma55</font></i></a>, <a class="ref" href="finseq_6.html#D1">FINSEQ_6:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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