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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">q1</font> be   <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>, <font color="Maroon">q2</font> be   <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/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green">E42</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:25"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">q1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="jordan7.html#T5">JORDAN7:5</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1"/>
( <font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> or ( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> ) or ( <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><b>suppose </b></a><a NAME="E1:25_1_1"/>
<font color="Maroon">q2</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:25_1_1"/>
 <a href="jordan7.html#K1">Segment</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">C</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_2"><b>suppose </b></a><a NAME="E1:25_1_2"/>
( <font color="Maroon">q1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><a NAME="E2:25_1_2"/><b>then </b>
 <a href="jordan7.html#K1">Segment</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">q1</font></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i>, <a class="ref" href="jordan7.html#T8">JORDAN7:8</a></i>;<br/><b>hence </b><a NAME="E3:25_1_2"/>
 <a href="jordan7.html#K1">Segment</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">C</font> is <a href="compts_1.html#V6">compact</a>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_3"><b>suppose </b></a><a NAME="E1:25_1_3"/>
( <font color="Maroon">q1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q2</font> &amp; <font color="Maroon">q2</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><a NAME="E2:25_1_3"/><b>then </b>
 <a href="jordan7.html#K1">Segment</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">C</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E3:25_1_3"/>
 <a href="jordan7.html#K1">Segment</a> <font color="Maroon">q1</font>,<font color="Maroon">q2</font>,<font color="Maroon">C</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="ref" href="jordan5a.html#T1">JORDAN5A:1</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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