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<div class="add">

<b>let </b><font color="Maroon">p1</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">p2</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">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</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="E1:52"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">P</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> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">p</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 </span>}</span>  &amp; ( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font> )
 ;<br/>

<a NAME="E2:52"/><b>then </b><i><font color="Green">E31</font></i>: 
<font color="Maroon">P</font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a> 
 
<b>by </b><i><a class="ref" href="jgraph_3.html#T36">JGRAPH_3:36</a></i>;<br/>
<a NAME="E3:52"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/>
<a NAME="E4:52"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">P</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> : <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">p</font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 </span>}</span>  &amp; <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; ( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font> )
 
<b>by </b><i>, <a class="ref" href="jgraph_5.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan7.html#T5">JORDAN7:5</a></i>;<br/>
<a NAME="E5:52"/><i><font color="Green">E34</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</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> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) </span>}</span> 
 
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E4:52"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E6:52"/><i><font color="Green">E45</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</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> : ( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 ) </span>}</span> 
 
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E5:52"><i><font color="Green">E34</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1"/>
( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or ( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_1"><b>suppose </b></a><a NAME="E1:52_1_1"/>
<font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 ;<br/><div class="add"><a NAME="E2:52_1_1"/><b>then </b>
<font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="jgraph_5.html#T8" target="_self">Th8</a>, </i>;<br/><a NAME="E3:52_1_1"/><b>then </b>
<font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="txt" href="jgraph_5.html#E6:52"><i><font color="Green">E45</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">p8</font> being   <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><b> such that </b><br/><a NAME="E5:52_1_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">p8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">p8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E6:52_1_1"/>
( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2"><b>suppose </b></a><a NAME="E1:52_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">p2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><div class="add"><a NAME="E2:52_1_2"/><b>then </b>
for <font color="Olive">p8</font> being  <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>  holds <br/>(  not <font color="Olive">p8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> or  not <font color="Olive">p8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> or  not <font color="Olive">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/><a NAME="E3:52_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
not <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:52_1_2_1_1"/>
( ( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) or ( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) )
 </span><b>by </b><i>, , <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_1"><b>case </b></a><a NAME="E1:52_1_2_1_1_1"/>
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p8</font> being   <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><b> such that </b><br/><a NAME="E3:52_1_2_1_1_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">p8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">p8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i/>;<br/><b>thus </b><a NAME="E4:52_1_2_1_1_1"/>
( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="52_1_2_1_1_2"><b>case </b></a><a NAME="E1:52_1_2_1_1_2"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p2</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">p8</font> being   <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><b> such that </b><br/><a NAME="E3:52_1_2_1_1_2"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">p8</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">p8</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p8</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a> 0 )
 <b>by </b><i/>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_2_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:52_1_2_1_1_2_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:52_1_2_1_1_2_1"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p2</font>,<font color="Maroon">p1</font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E3:52_1_2_1_1_2_1"/>
contradiction
 <b>by </b><i>, , <a class="ref" href="jgraph_5.html#T10" target="_self">Th10</a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div><a NAME="E5:52_1_2_1_1_2"/><b>then </b>
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 
 <b>by </b><i>, , <a class="ref" href="jgraph_5.html#T9" target="_self">Th9</a>, </i>;<br/><b>hence </b><a NAME="E6:52_1_2_1_1_2"/>
( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 <b>by </b><i/>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:52_1_2"/>
( <font color="Maroon">p1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 or <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
