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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/>

<a NAME="E1:39"/><i><font color="Green" title="E35">A1</font></i>: 
 <a href="jordan19.html#K4" title="JORDAN19:func.4">South_Arc</a> <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="jordan22.html#T38" target="_self" title="JORDAN22:th.38">Th38</a></i>;<br/>
<a NAME="E2:39"/><i><font color="Green" title="E36">A2</font></i>: 
 <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan19.html#K4" title="JORDAN19:func.4">South_Arc</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="jordan22.html#T36" target="_self" title="JORDAN22:th.36">Th36</a></i>;<br/>
<a NAME="E3:39"/><i><font color="Green" title="E37">A3</font></i>: 
 <a href="jordan19.html#K3" title="JORDAN19:func.3">North_Arc</a> <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="jordan22.html#T37" target="_self" title="JORDAN22:th.37">Th37</a></i>;<br/>
<a NAME="E4:39"/><i><font color="Green" title="E38">A4</font></i>: 
 <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan19.html#K3" title="JORDAN19:func.3">North_Arc</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="jordan22.html#T35" target="_self" title="JORDAN22:th.35">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" title="JORDAN6:pred.1">LE</a>  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font>, <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font> or  <a href="jordan6.html#R1" title="JORDAN6:pred.1">LE</a>  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">C</font>, <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font> )
 </span><b>by </b><i><a class="txt" href="jordan22.html#E1:39"><i><font color="Green" title="E35">A1</font></i></a>, <a class="txt" href="jordan22.html#E2:39"><i><font color="Green" title="E36">A2</font></i></a>, <a class="txt" href="jordan22.html#E3:39"><i><font color="Green" title="E37">A3</font></i></a>, <a class="txt" href="jordan22.html#E4:39"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="jordan16.html#T11" title="JORDAN16:th.11">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" title="JORDAN6:pred.1">LE</a>  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font>, <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font>
 ;<br/><div class="add"><a NAME="E2:39_1_1_1"/><b>then </b>
( (  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font> ) )
 <b>by </b><i><a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/><b>hence </b><a NAME="E3:39_1_1_1"/>
(  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="ref" href="jordan21.html#T62" title="JORDAN21:th.62">JORDAN21:62</a>, <a class="ref" href="jordan21.html#T63" title="JORDAN21:th.63">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" title="JORDAN6:pred.1">LE</a>  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">C</font>, <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c1">C</font>
 ;<br/><div class="add"><a NAME="E2:39_1_1_2"/><b>then </b>
( (  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font> ) )
 <b>by </b><i><a class="ref" href="jordan6.html#D10" title="JORDAN6:def.10">JORDAN6:def 10</a></i>;<br/><b>hence </b><a NAME="E3:39_1_1_2"/>
(  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font> )
 <b>by </b><i><a class="ref" href="jordan21.html#T62" title="JORDAN21:th.62">JORDAN21:62</a>, <a class="ref" href="jordan21.html#T63" title="JORDAN21:th.63">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" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font> ) or (  <a href="jordan21.html#K1" title="JORDAN21:func.1">UMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font> &amp;  <a href="jordan21.html#K2" title="JORDAN21:func.2">LMP</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K8" title="JORDAN6:func.8">Upper_Arc</a> <font color="Maroon" title="c1">C</font> ) )
 ;<br/>


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