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<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">S</font> be    <a href="jordan_a.html#M1" target="_self">Segmentation</a> of <font color="Maroon">C</font>;<br/><b>let </b><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="E1:34"/><i><font color="Green">E42</font></i>: 
1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">j</font>
 <b>and </b><br/><a NAME="E2:34"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span> <a href="binarith.html#K3">-'</a> 1
 ;<br/>

<a NAME="E3:34"/><i><font color="Green">E44</font></i>: 
 <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="jordan7.html#K1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> 1</span>)</span>,<font color="Maroon">C</font>
 
<b>by </b><i/>;<br/>
<a NAME="E4:34"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="sprect_3.html#T6">SPRECT_3:6</a></i>;<br/>
<a NAME="E5:34"/>
<font color="Maroon">j</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="jordan3.html#T7">JORDAN3:7</a></i>;<br/>
<a NAME="E6:34"/><b>then </b>
 <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">S</font>,<font color="Maroon">j</font> <a href="hidden.html#R1">=</a>  <a href="jordan7.html#K1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="finseq_4.html#K4">/.</a> <span class="p2">(<span class="default"><font color="Maroon">j</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">C</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:34"/>
 <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font></span>)</span> <a href="subset_1.html#R1">misses</a>  <a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">S</font>,<font color="Maroon">j</font>
 <b>by </b><i>, , , </i>;<br/>


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