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<b>let </b><font color="Maroon">C</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/>

<b>set </b><font color="Maroon">X</font> = <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>,<font color="Olive">p</font>,<font color="Maroon">C</font> or (  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> ) ) </span>}</span> ;<br/>
<a NAME="E1:23"/><i><font color="Green">E42</font></i>: 
 <a href="jordan7.html#K1">Segment</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : (  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>,<font color="Olive">p</font>,<font color="Maroon">C</font> or (  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan7.html#D1">JORDAN7:def 1</a></i>;<br/>
<b>thus </b><a NAME="E2:23"/>
<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a>  <a href="jordan7.html#K1">Segment</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">C</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:23_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">e</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E3:23_1"/>
 <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>,<font color="Maroon">p</font>,<font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="jordan7.html#T3">JORDAN7:3</a></i>;<br/>
<b>hence </b><a NAME="E4:23_1"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a>  <a href="jordan7.html#K1">Segment</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">C</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:23"/>
 <a href="jordan7.html#K1">Segment</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font></span>)</span>,<font color="Maroon">C</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan16.html#T10">JORDAN16:10</a></i>;<br/>


</div>
