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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon" title="c2">S</font> be    <a href="jordan_a.html#M1" title="JORDAN_A:mode.1">Segmentation</a> of <font color="Maroon" title="c1">C</font>;<br/><b>let </b><font color="Maroon" title="c3">j</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>assume </b><b>that </b><br/><a NAME="E1:33"/><i><font color="Green" title="E26">A1</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">j</font>
 <b>and </b><br/><a NAME="E2:33"/><i><font color="Green" title="E27">A2</font></i>: 
<font color="Maroon" title="c3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1
 ;<br/>

<a NAME="E3:33"/><i><font color="Green" title="E28">A3</font></i>: 
 <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> 1</span>)</span>,<font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="jordan_a.html#D4" target="_self" title="JORDAN_A:def.4">Def4</a></i>;<br/>
<a NAME="E4:33"/><i><font color="Green" title="E29">A4</font></i>: 
<font color="Maroon" title="c3">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E2:33"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="binarith.html#T71" title="BINARITH:th.71">BINARITH:71</a></i>;<br/>
<a NAME="E5:33"/>
<font color="Maroon" title="c3">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E2:33"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="binarith.html#T62" title="BINARITH:th.62">BINARITH:62</a></i>;<br/>
<a NAME="E6:33"/><b>then </b>
 <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">j</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan7.html#K1" title="JORDAN7:func.1">Segment</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c3">j</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E1:33"><i><font color="Green" title="E26">A1</font></i></a>, <a class="ref" href="jordan_a.html#D4" target="_self" title="JORDAN_A:def.4">Def4</a></i>;<br/>
<b>hence </b><a NAME="E7:33"/>
 <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">j</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E1:33"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="jordan_a.html#E3:33"><i><font color="Green" title="E28">A3</font></i></a>, <a class="txt" href="jordan_a.html#E4:33"><i><font color="Green" title="E29">A4</font></i></a>, <a class="ref" href="jordan_a.html#D3" target="_self" title="JORDAN_A:def.3">Def3</a></i>;<br/>


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