<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/>

<a NAME="E1:39"/><i><font color="Green">E40</font></i>: 
 <a href="jordan19.html#K4">South_Arc</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="jordan22.html#T38" target="_self">Th38</a></i>;<br/>
<a NAME="E2:39"/><i><font color="Green">E41</font></i>: 
 <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan19.html#K4">South_Arc</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="jordan22.html#T36" target="_self">Th36</a></i>;<br/>
<a NAME="E3:39"/><i><font color="Green">E42</font></i>: 
 <a href="jordan19.html#K3">North_Arc</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="jordan22.html#T37" target="_self">Th37</a></i>;<br/>
<a NAME="E4:39"/><i><font color="Green">E43</font></i>: 
 <a href="jordan21.html#K1">UMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan19.html#K3">North_Arc</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="jordan22.html#T35" target="_self">Th35</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:39_1_1"/>
(  <a href="jordan6.html#R1">LE</a>  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>1</sub></font>, <a href="jordan21.html#K1">UMP</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> or  <a href="jordan6.html#R1">LE</a>  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">c<sub>1</sub></font>, <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="jordan22.html#E1:39"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan22.html#E2:39"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan22.html#E3:39"><i><font color="Green">E42</font></i></a>, <a class="txt" href="jordan22.html#E4:39"><i><font color="Green">E43</font></i></a>, <a class="ref" href="jordan16.html#T11">JORDAN16:11</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_1"><b>case </b></a><a NAME="E1:39_1_1_1"/>
 <a href="jordan6.html#R1">LE</a>  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>1</sub></font>, <a href="jordan21.html#K1">UMP</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:39_1_1_1"/><b>then </b>
( (  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font> ) or (  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font> ) or (  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/><b>hence </b><a NAME="E3:39_1_1_1"/>
(  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="jordan21.html#T62">JORDAN21:62</a>, <a class="ref" href="jordan21.html#T63">JORDAN21:63</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="39_1_1_2"><b>case </b></a><a NAME="E1:39_1_1_2"/>
 <a href="jordan6.html#R1">LE</a>  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">c<sub>1</sub></font>, <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:39_1_1_2"/><b>then </b>
( (  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font> ) or (  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font> ) or (  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/><b>hence </b><a NAME="E3:39_1_1_2"/>
(  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="jordan21.html#T62">JORDAN21:62</a>, <a class="ref" href="jordan21.html#T63">JORDAN21:63</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E6:39"/>
( (  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> ) or (  <a href="jordan21.html#K1">UMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="jordan21.html#K2">LMP</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> ) )
 ;<br/>


</div>
