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

<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">p</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>let </b><font color="Maroon">q</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><b>that </b><br/><a NAME="E1:10"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:10"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E3:10"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">C</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="jordan6.html#T65">JORDAN6:65</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:10_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> &amp; <font color="Maroon">p</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> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> &amp; <font color="Maroon">q</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> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> &amp; <font color="Maroon">p</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> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> &amp; <font color="Maroon">q</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">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font> ) )
 </span><b>by </b><i>, , , <a class="ref" href="jordan7.html#T1">JORDAN7:1</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_1"><b>suppose </b></a><a NAME="E1:10_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_1"/>
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">C</font> or  <a href="jordan6.html#R1">LE</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>,<font color="Maroon">C</font> )
 <b>by </b><i>, <a class="ref" href="jordan6.html#T71">JORDAN6:71</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:10_1_2"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> &amp; <font color="Maroon">p</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> )
 <b>and </b><br/><a NAME="E2:10_1_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> &amp; <font color="Maroon">q</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> )
 <b>and </b><br/><a NAME="E3:10_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 ;<br/><div class="add"><a NAME="E4:10_1_2"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T65">JORDAN6:65</a></i>;<br/><a NAME="E5:10_1_2"/><b>then </b>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> or  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , <a class="ref" href="jordan5c.html#T14">JORDAN5C:14</a></i>;<br/><b>hence </b><a NAME="E6:10_1_2"/>
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">C</font> or  <a href="jordan6.html#R1">LE</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>,<font color="Maroon">C</font> )
 <b>by </b><i>, , <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_3"><b>suppose </b></a><a NAME="E1:10_1_3"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> &amp; <font color="Maroon">p</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> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:10_1_3"/>
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">C</font> or  <a href="jordan6.html#R1">LE</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>,<font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_4"><b>suppose </b></a><a NAME="E1:10_1_4"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">C</font> &amp; <font color="Maroon">q</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"><b>hence </b><a NAME="E2:10_1_4"/>
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">C</font> or  <a href="jordan6.html#R1">LE</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>,<font color="Maroon">C</font> )
 <b>by </b><i><a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="10_1_5"><b>suppose </b></a><b>that </b><a NAME="E1:10_1_5"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E2:10_1_5"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>
 <b>and </b><br/><a NAME="E3:10_1_5"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 ;<br/><div class="add"><a NAME="E4:10_1_5"/>
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T65">JORDAN6:65</a></i>;<br/><a NAME="E5:10_1_5"/><b>then </b>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font> or  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">C</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">C</font> )
 <b>by </b><i>, , , <a class="ref" href="jordan5c.html#T14">JORDAN5C:14</a></i>;<br/><b>hence </b><a NAME="E6:10_1_5"/>
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">C</font> or  <a href="jordan6.html#R1">LE</a> <font color="Maroon">q</font>,<font color="Maroon">p</font>,<font color="Maroon">C</font> )
 <b>by </b><i>, , <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
