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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">e</font> be   <a href="real_1.html#NM1">Real</a>;<br/>





<b>assume </b><a NAME="E1:44"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds  <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="Olive">i</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 ;<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="E3:44"/><i><font color="Green">E43</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">i</font></span>)</span></span>)</span> where <font color="Olive">i</font> is    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  : <font color="Olive">i</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="E4:44"/><i><font color="Green">E44</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/>
<a NAME="E5:44"/>
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font> holds <br/><font color="Olive">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E1:44_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S1</font>
 ;<br/>

<a NAME="E2:44_1"/><b>then </b>
 ex <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <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="Olive">i</font></span>)</span> &amp; <font color="Olive">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/>;<br/>
<b>hence </b><a NAME="E3:44_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:44"/>
 <a href="jordan_a.html#K4" target="_self">diameter</a> <font color="Maroon">S</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">e</font>
 <b>by </b><i>, <a class="ref" href="jordan16.html#T1">JORDAN16:1</a></i>;<br/>


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