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<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>, <font color="Maroon">p3</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">p4</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="E1:67"/><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">p4</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p3</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font> )
 ;<br/>

<a NAME="E2:67"/><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="E3:67"/><b>then </b><i><font color="Green">E32</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#D8">JORDAN6:def 8</a></i>;<br/>
<a NAME="E4:67"/><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">p3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">p4</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p2</font>,<font color="Maroon">p3</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p3</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font> )
 
<b>by </b><i>, , <a class="ref" href="jordan7.html#T5">JORDAN7:5</a></i>;<br/>
<a NAME="E5:67"/><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">p7</font> where <font color="Olive">p7</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">p7</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p7</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#E4:67"><i><font color="Green">E33</font></i></a></i>;<br/>
<a NAME="E6:67"/>
 <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>,0</span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:67"/><b>then </b><i><font color="Green">E45</font></i>: 
( <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> 1 &amp; <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E8:67"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">p4</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#T11" target="_self">Th11</a>, </i>;<br/>
<a NAME="E9:67"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">p3</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#E7:67"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E10:67"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p4</font>,<font color="Maroon">p3</font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 
<b>by </b><i>, , <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/>
<a NAME="E11:67"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">p4</font>,<font color="Maroon">p3</font>,<font color="Maroon">P</font>
 
<b>by </b><i>, , <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E12:67"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">p3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font>
 
<b>by </b><i>, , <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/>
<a NAME="E13:67"/><i><font color="Green">E49</font></i>: 
<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="txt" href="jgraph_5.html#E7:67"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E14:67"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p4</font>,<font color="Maroon">p2</font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 
<b>by </b><i>, , <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/>
<a NAME="E15:67"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">p4</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font>
 
<b>by </b><i>, <a class="txt" href="jgraph_5.html#E15:67"><i><font color="Green">E73</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E16:67"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">p2</font>,<font color="Maroon">p4</font>,<font color="Maroon">P</font>
 
<b>by </b><i>, , <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/>
<a NAME="E17:67"/><b>then </b><i><font color="Green">E75</font></i>: 
<font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font>
 
<b>by </b><i>, , <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/>
<a NAME="E18:67"/><i><font color="Green">E77</font></i>: 
<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#E15:67"><i><font color="Green">E73</font></i></a>, <a class="txt" href="jgraph_5.html#E7:67"><i><font color="Green">E45</font></i></a></i>;<br/>
<a NAME="E19:67"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p4</font>,<font color="Maroon">p1</font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 
<b>by </b><i>, , <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/>
<a NAME="E20:67"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">p4</font>,<font color="Maroon">p1</font>,<font color="Maroon">P</font>
 
<b>by </b><i>, , <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<a NAME="E21:67"/><b>then </b>
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p4</font>
 
<b>by </b><i>, , <a class="txt" href="jgraph_5.html#E17:67"><i><font color="Green">E75</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/>
<b>hence </b><a NAME="E22:67"/>
 ex <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ex <font color="Olive">q1</font>, <font color="Olive">q2</font>, <font color="Olive">q3</font>, <font color="Olive">q4</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> st <br/>( <font color="Olive">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; ( for <font color="Olive">q</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  <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Olive">q</font></span><a href="toprns_1.html#K5">.|</a></span> ) &amp; <font color="Olive">q1</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font> &amp; <font color="Olive">q2</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font> &amp; <font color="Olive">q3</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p3</font> &amp; <font color="Olive">q4</font> <a href="hidden.html#R1">=</a> <font color="Olive">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p4</font> &amp; <font color="Olive">q1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">q1</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">q2</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">q2</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">q3</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">q3</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">q4</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp; <font color="Olive">q4</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">q1</font>,<font color="Olive">q2</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">q2</font>,<font color="Olive">q3</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">q3</font>,<font color="Olive">q4</font>,<font color="Maroon">P</font> )
 <b>by </b><i>, , , <a class="txt" href="jgraph_5.html#E17:67"><i><font color="Green">E75</font></i></a>, </i>;<br/>


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