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<b>let </b><font color="Maroon">P</font> be   <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="E1:61"/>
<font color="Maroon">P</font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a> 
 ;<br/>

<a NAME="E2:61"/><b>then </b><i><font color="Green">E37</font></i>: 
( <span class="p1">(<span class="default"><a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> )
 
<b>by </b><i><a class="ref" href="jordan6.html#T4" target="_self">Th4</a></i>;<br/>
<b>set </b><font color="Maroon">B</font> =  <a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font>;<br/>
<b>set </b><font color="Maroon">A</font> =  <a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font>;<br/>
<i><font color="Green">E47</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>
<br/>.= 
 <a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T51">XBOOLE_1:51</a></i>
;<br/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p3">(<span class="default"><a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T40">XBOOLE_1:40</a></i>
<br/>.= 
 <a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font>
<b>by </b><i><a class="ref" href="xboole_1.html#T51">XBOOLE_1:51</a></i>
;<br/>
<b>hence </b><a NAME="E5:61"/>
(  <a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span> &amp;  <a href="jordan6.html#K8" target="_self">Upper_Arc</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9" target="_self">Lower_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i>, <a class="txt" href="jordan6.html#E3:61"><i><font color="Green">E47</font></i></a></i>;<br/>


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