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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">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>





<b>consider </b><font color="Maroon">S1</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E2:43"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S1</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan_a.html#K3" target="_self">diameter</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">S</font>,<font color="Olive">j</font></span>)</span></span>)</span> where <font color="Olive">j</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">j</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S</font> </span>}</span> 
 <b>and </b><br/><a NAME="E3:43"/><i><font color="Green">E43</font></i>: 
 <a href="jordan_a.html#K4" target="_self">diameter</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="pre_circ.html#K1">max</a> <font color="Maroon">S1</font>
 <b>by </b><i/>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:43_1"/>
( ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> ) or  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> or  not <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_1"><b>suppose </b></a><a NAME="E1:43_1_1"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> )
 ;<br/><div class="add"><a NAME="E2:43_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:43_1_1"/><b>then </b>
 <a href="jordan_a.html#K3" target="_self">diameter</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">S</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:43_1_1"/>
 <a href="jordan_a.html#K3" target="_self">diameter</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">S</font>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan_a.html#K4" target="_self">diameter</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="43_1_2"><b>suppose </b></a><a NAME="E1:43_1_2"/><i><font color="Green">E44</font></i>: 
(  not 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> or  not <font color="Maroon">i</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> )
 ;<br/><div class="add"><i><font color="Green">E45</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="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 href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">S</font>,<font color="Maroon">i</font>
<b>by </b><i>, </i>
;<br/><a NAME="E3:43_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">S</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6">FINSEQ_5:6</a></i>;<br/><a NAME="E4:43_1_2"/><b>then </b>
 <a href="jordan_a.html#K3" target="_self">diameter</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">S</font>,<font color="Maroon">i</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E5:43_1_2"/>
 <a href="jordan_a.html#K3" target="_self">diameter</a> <span class="p1">(<span class="default"><a href="jordan_a.html#K2" target="_self">Segm</a> <font color="Maroon">S</font>,<font color="Maroon">i</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan_a.html#K4" target="_self">diameter</a> <font color="Maroon">S</font>
 <b>by </b><i>, <a class="ref" href="pre_circ.html#D1">PRE_CIRC:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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