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

<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="E1:8"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">P</font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a> 
 ;<br/>

<a NAME="E2:8"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="topreal5.html#T25">TOPREAL5:25</a></i>;<br/>
<a NAME="E3:8"/><i><font color="Green">E30</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="E4:8"/><i><font color="Green">E36</font></i>: 
<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> : (  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>,<font color="Olive">p</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">p</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Maroon">P</font> ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<a NAME="E1:8_1"/><i><font color="Green">E37</font></i>: 
<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> : (  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>,<font color="Olive">p</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">p</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Maroon">P</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:8_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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> : (  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>,<font color="Olive">p</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">p</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Maroon">P</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p</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="E3:8_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>,<font color="Maroon">p</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">p</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Maroon">P</font> )
 ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1_1_1"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</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">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</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> ) or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp;  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</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">p</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</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="8_1_1_1_1"><b>suppose </b></a><a NAME="E1:8_1_1_1_1"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</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>hence </b><a NAME="E2:8_1_1_1_1"/>
<font color="Maroon">x</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><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_2"/>
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:8_1_1_1_2"/>
<font color="Maroon">x</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><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_3"><b>suppose </b></a><a NAME="E1:8_1_1_1_3"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp;  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</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">p</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</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"><a NAME="E2:8_1_1_1_3"/>
 <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="E3:8_1_1_1_3"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="jordan6.html#T69">JORDAN6:69</a></i>;<br/><b>hence </b><a NAME="E4:8_1_1_1_3"/>
<font color="Maroon">x</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="topreal1.html#T4">TOPREAL1:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:8_1"/>
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</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> : (  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>,<font color="Olive">p</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">p</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Maroon">P</font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:8_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">x</font> as   <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/>
<a NAME="E3:8_1_2"/><i><font color="Green">E47</font></i>: 
(  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp;  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp;  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</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>, </i>;<br/>
<a NAME="E4:8_1_2"/>
 <a href="jordan5c.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</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="E5:8_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
 <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>,<font color="Maroon">p2</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="E6:8_1_2"/>
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">p2</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Maroon">P</font>
 
<b>by </b><i>, , , <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<b>hence </b><a NAME="E7:8_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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> : (  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>,<font color="Olive">p</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">p</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Maroon">P</font> ) </span>}</span> 
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:8_1"/>
<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> : (  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>,<font color="Olive">p</font>,<font color="Maroon">P</font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Olive">p</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Maroon">P</font> ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:8"/>
<span class="p1">{<span class="default"> <font color="Olive">p1</font> where <font color="Olive">p1</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> : (  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Olive">p1</font>,<font color="Maroon">P</font> or (  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2"><b>proof </b></a><div class="add">

<a NAME="E1:8_2"/><i><font color="Green">E49</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">p1</font> where <font color="Olive">p1</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> : (  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Olive">p1</font>,<font color="Maroon">P</font> or (  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:8_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p1</font> where <font color="Olive">p1</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> : (  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Olive">p1</font>,<font color="Maroon">P</font> or (  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">p1</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="E3:8_2_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; (  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">P</font> or (  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <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/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_2_1_1"/>
(  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">P</font> or (  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <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> ) )
 </span><b>by </b><i/>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1_1_1"><b>suppose </b></a><a NAME="E1:8_2_1_1_1"/><i><font color="Green">E51</font></i>: 
 <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Maroon">p1</font>,<font color="Maroon">P</font>
 ;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_2_1_1_1_1"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> or (  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</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">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</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> ) )
 </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="8_2_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:8_2_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:8_2_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:8_2_1_1_1_1_2"/><i><font color="Green">E52</font></i>: 
(  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</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">p1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="jordan5c.html#R1">LE</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</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> )
 ;<br/><div class="add"><a NAME="E2:8_2_1_1_1_1_2"/><i><font color="Green">E53</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#T65">JORDAN6:65</a></i>;<br/><a NAME="E3:8_2_1_1_1_1_2"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">p1</font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</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="E4:8_2_1_1_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="jordan5c.html#T12">JORDAN5C:12</a></i>;<br/><a NAME="E5:8_2_1_1_1_1_2"/>
 <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/><b>hence </b><a NAME="E6:8_2_1_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_2_1_1_2"><b>suppose </b></a><a NAME="E1:8_2_1_1_2"/>
(  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <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/><div class="add"><a NAME="E2:8_2_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</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="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E3:8_2_1_1_2"/><b>then </b>
<font color="Maroon">x</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">P</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">P</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><b>hence </b><a NAME="E4:8_2_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:8_2"/>
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">p1</font> where <font color="Olive">p1</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> : (  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Olive">p1</font>,<font color="Maroon">P</font> or (  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</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="E1:8_2_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">p2</font> = <font color="Maroon">x</font> as   <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/>
<a NAME="E3:8_2_2"/>
 <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#T65">JORDAN6:65</a></i>;<br/>
<a NAME="E4:8_2_2"/><b>then </b>
( (  not  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> or  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> ) implies (  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</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> ) )
 
<b>by </b><i>, <a class="ref" href="sprect_1.html#T16">SPRECT_1:16</a>, <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<a NAME="E5:8_2_2"/><b>then </b>
(  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Maroon">p2</font>,<font color="Maroon">P</font> or (  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <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> ) )
 
<b>by </b><i><a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/>
<b>hence </b><a NAME="E6:8_2_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">p1</font> where <font color="Olive">p1</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> : (  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Olive">p1</font>,<font color="Maroon">P</font> or (  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:8_2"/>
<span class="p1">{<span class="default"> <font color="Olive">p1</font> where <font color="Olive">p1</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> : (  <a href="jordan6.html#R1">LE</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>,<font color="Olive">p1</font>,<font color="Maroon">P</font> or (  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">p1</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font> ) ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:8"/>
(  <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">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font> &amp;  <a href="jordan7.html#K1" target="_self">Segment</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font></span>)</span>,<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font> )
 <b>by </b><i>, , </i>;<br/>


</div>
