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

<b>let </b><font color="Maroon">c<sub>1</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>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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>assume </b><a NAME="E1:16"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a>  &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:16"/><b>then </b><i><font color="Green">E15</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="jordan7.html#T5" target="_self">Th5</a></i>;<br/>
<b>thus </b><a NAME="E3:16"/>
<span class="p1">(<span class="default"><a href="jordan7.html#K1" target="_self">Segment</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" target="_self">Segment</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:16_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" target="_self">Segment</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" target="_self">Segment</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<a NAME="E2:16_1"/><b>then </b><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E3:16_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</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> : (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font> ) ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan7.html#D1" target="_self">Def1</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>5</sub></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="E5:16_1"/><i><font color="Green">E17</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font> ) ) )
 ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:16_1_1_1"/>
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font> or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font> ) )
 </span><b>by </b><i><a class="txt" href="jordan7.html#E5:16_1"><i><font color="Green">E17</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_1"><b>case </b></a><a NAME="E1:16_1_1_1_1"/><i><font color="Green">E18</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:16_1_1_1_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</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> : (  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan7.html#E1:16"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan7.html#E2:16_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="jordan7.html#D1" target="_self">Def1</a></i>;<br/><a NAME="E3:16_1_1_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 ;<br/><a NAME="E4:16_1_1_1_1"/><b>then </b>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan7.html#E1:16"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan7.html#E5:16_1"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan7.html#E1:16_1_1_1_1"><i><font color="Green">E18</font></i></a>, <a class="ref" href="jordan6.html#T73">JORDAN6:73</a></i>;<br/><b>hence </b><a NAME="E5:16_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan7.html#E1:16"><i><font color="Green">E14</font></i></a>, <a class="ref" href="jordan6.html#T72">JORDAN6:72</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="16_1_1_1_2"><b>case </b></a><a NAME="E1:16_1_1_1_2"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:16_1_1_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan7.html#E5:16_1"><i><font color="Green">E17</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:16_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:16"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E5:16"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:16"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</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> : (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> or ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font> ) ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan7.html#E2:16"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E7:16"/><b>then </b><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="jordan7.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E8:16"/>
 <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:16"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan7.html#E2:16"><i><font color="Green">E15</font></i></a>, <a class="ref" href="jordan7.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E9:16"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan7.html#E1:16"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan7.html#E5:16"><i><font color="Green">E16</font></i></a>, <a class="ref" href="jordan7.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E10:16"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" target="_self">Segment</a> <font color="Maroon">c<sub>3</sub></font>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan7.html#K1" target="_self">Segment</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan7.html#E7:16"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>


</div>
