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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">i</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>consider </b><font color="Maroon" title="c4">S1</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E2:42"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c4">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="jordan_a.html#K3" title="JORDAN_A:func.3">diameter</a> <span class="p3">(<span class="default"><a href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Olive" title="b1">j</font></span>)</span></span>)</span> where <font color="Olive" title="b1">j</font> is    <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>  : <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">S</font> </span>}</span> 
 <b>and </b><br/><a NAME="E3:42"/><i><font color="Green" title="E35">A2</font></i>: 
 <a href="jordan_a.html#K4" title="JORDAN_A:func.4">diameter</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">max</a> <font color="Maroon" title="c4">S1</font>
 <b>by </b><i><a class="ref" href="jordan_a.html#D6" target="_self" title="JORDAN_A:def.6">Def6</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1"/>
( ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</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> ) or  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> or  not <font color="Maroon" title="c3">i</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><b>suppose </b></a><a NAME="E1:42_1_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> &amp; <font color="Maroon" title="c3">i</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> )
 ;<br/><div class="add"><a NAME="E2:42_1_1"/><b>then </b>
<font color="Maroon" title="c3">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><a NAME="E3:42_1_1"/><b>then </b>
 <a href="jordan_a.html#K3" title="JORDAN_A:func.3">diameter</a> <span class="p1">(<span class="default"><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">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S1</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E2:42"><i><font color="Green" title="E34">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E4:42_1_1"/>
 <a href="jordan_a.html#K3" title="JORDAN_A:func.3">diameter</a> <span class="p1">(<span class="default"><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">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan_a.html#K4" title="JORDAN_A:func.4">diameter</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E3:42"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2"><b>suppose </b></a><a NAME="E1:42_1_2"/><i><font color="Green" title="E36">A3</font></i>: 
(  not 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">i</font> or  not <font color="Maroon" title="c3">i</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> )
 ;<br/><div class="add"><i><font color="Green" title="E37">A4</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="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 href="jordan_a.html#K2" title="JORDAN_A:func.2">Segm</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">i</font>
<b>by </b><i><a class="txt" href="jordan_a.html#E1:42_1_2"><i><font color="Green" title="E36">A3</font></i></a>, <a class="ref" href="jordan_a.html#D4" target="_self" title="JORDAN_A:def.4">Def4</a></i>
;<br/><a NAME="E3:42_1_2"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="ref" href="finseq_5.html#T6" title="FINSEQ_5:th.6">FINSEQ_5:6</a></i>;<br/><a NAME="E4:42_1_2"/><b>then </b>
 <a href="jordan_a.html#K3" title="JORDAN_A:func.3">diameter</a> <span class="p1">(<span class="default"><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">i</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">S1</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E2:42"><i><font color="Green" title="E34">A1</font></i></a>, <a class="txt" href="jordan_a.html#E2:42_1_2"><i><font color="Green" title="E37">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E5:42_1_2"/>
 <a href="jordan_a.html#K3" title="JORDAN_A:func.3">diameter</a> <span class="p1">(<span class="default"><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">i</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan_a.html#K4" title="JORDAN_A:func.4">diameter</a> <font color="Maroon" title="c2">S</font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E3:42"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="pre_circ.html#D1" title="PRE_CIRC:def.1">PRE_CIRC:def 1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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