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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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">c<sub>3</sub></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="E1:48"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B 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">b<sub>1</sub></font></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 1 </span>}</span>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<a NAME="E2:48"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>3</sub></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="E3:48"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>3</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>3</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>3</sub></font>;<br/>
<a NAME="E4:48"/><i><font color="Green">E49</font></i>: 
(  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>3</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>3</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>3</sub></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">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span><a href="tarski.html#K2">}</a></span> &amp; <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="jordan5c.html#K1">First_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <span class="p1">(<span class="default"><a href="jordan5c.html#K2">Last_Point</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="jordan6.html#K6">Vertical_Line</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="pscomp_1.html#K15">W-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p5">(<span class="default"><a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> 2</span>)</span></span>)</span></span>)</span> <a href="euclid.html#K22">`2</a>  )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E2:48"><i><font color="Green">E47</font></i></a>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/>
<a NAME="E5:48"/><b>then </b>
 <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E6:48"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:48"/><i><font color="Green">E51</font></i>: 
( ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>3</sub></font> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>3</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>3</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>3</sub></font> ) or ( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>3</sub></font> &amp; not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>3</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>3</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>3</sub></font> ) )
 
<b>by </b><i><a class="txt" href="jgraph_5.html#E1:48"><i><font color="Green">E46</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:48_1"/><i><font color="Green">E52</font></i>: 
not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:48_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E1:48"><i><font color="Green">E46</font></i></a>, <a class="txt" href="jgraph_5.html#E7:48"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:48_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E4:48"><i><font color="Green">E49</font></i></a>, <a class="txt" href="jgraph_5.html#E7:48"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jgraph_5.html#E1:48_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E4:48_1"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>3</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>3</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:48"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jgraph_5.html#E7:48"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jgraph_5.html#E1:48_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E5:48_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jgraph_5.html#E3:48"><i><font color="Green">E48</font></i></a>, <a class="txt" href="jgraph_5.html#E6:48"><i><font color="Green">E50</font></i></a>, <a class="txt" href="jgraph_5.html#E7:48"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jgraph_5.html#E1:48_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jgraph_5.html#E3:48_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:48"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


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