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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><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>set </b><font color="Maroon">c<sub>3</sub></font> =  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>assume </b><a NAME="E1:18"/><i><font color="Green">E14</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:18"/><i><font color="Green">E15</font></i>: 
<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>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan_a.html#E1:18"><i><font color="Green">E14</font></i></a>, <a class="ref" href="jordan17.html#T3">JORDAN17:3</a></i>;<br/>
<a NAME="E3:18"/><i><font color="Green">E16</font></i>: 
 <a href="pscomp_1.html#K27">W-min</a> <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>1</sub></font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T1">JORDAN7:1</a></i>;<br/>
<a NAME="E4:18"/><i><font color="Green">E17</font></i>: 
 <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</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>1</sub></font>, <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="jordan6.html#T65">JORDAN6:65</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <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>] means (  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> or ( <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">a<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> ) );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <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>] means  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <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>] means  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">a<sub>1</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <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>] means ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">a<sub>1</sub></font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">a<sub>1</sub></font>] );<br/>
<a NAME="E5:18"/><i><font color="Green">E18</font></i>: 
for <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> holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] iff <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</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>thus </b><a NAME="E1:18_1"/>
( ( not  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; not <!--HUMANRECFAILED-->( <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>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> ) ) or  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_1"/><i><font color="Green">E19</font></i>: 
(  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font> or ( <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>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> ) )
 ;<br/>

<div><i><font color="Green">E20</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <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="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><a NAME="E2:18_1_1_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"><a href="jordan6.html#K9">Lower_Arc</a> <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="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:18_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</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 class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K7">}</a></span>
 <b>by </b><i><a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><b>hence </b><a NAME="E4:18_1_1_1"/>
( <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> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_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> or <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font> or ( not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</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> ) or ( not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</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> ) )
 </span><b>by </b><i><a class="txt" href="jordan_a.html#E2:18_1_1"><i><font color="Green">E20</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_1"><b>suppose </b></a><a NAME="E1:18_1_1_2_1"/>
<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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_1_1_2_1"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>, <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="jordan_a.html#E3:18"><i><font color="Green">E16</font></i></a>, <a class="ref" href="jordan5c.html#T9">JORDAN5C:9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_2"><b>suppose </b></a><a NAME="E1:18_1_1_2_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_1_1_2_2"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>, <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="jordan_a.html#E3:18"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan_a.html#E4:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_3"><b>suppose </b></a><a NAME="E1:18_1_1_2_3"/>
( not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>4</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/><div class="add"><b>hence </b><a NAME="E2:18_1_1_2_3"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>, <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="jordan_a.html#E1:18_1_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_1_2_4"><b>suppose </b></a><b>that </b><a NAME="E1:18_1_1_2_4"/><i><font color="Green">E21</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:18_1_1_2_4"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</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/><div class="add"><a NAME="E3:18_1_1_2_4"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E1:18_1_1"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan_a.html#E2:18_1_1_2_4"><i><font color="Green">E22</font></i></a>, <a class="ref" href="jordan7.html#T5">JORDAN7:5</a></i>;<br/><a NAME="E4:18_1_1_2_4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan6.html#D9">JORDAN6:def 9</a></i>;<br/><a NAME="E5:18_1_1_2_4"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E1:18_1_1_2_4"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan_a.html#E3:18_1_1_2_4"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><b>hence </b><a NAME="E6:18_1_1_2_4"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>, <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="jordan_a.html#E4:18"><i><font color="Green">E17</font></i></a>, <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

<b>assume </b><a NAME="E2:18_1"/><i><font color="Green">E19</font></i>: 
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E3:18_1"/><i><font color="Green">E20</font></i>: 
 <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T76">JORDAN6:76</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_2"/>
(  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font> or  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>, <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="jordan_a.html#E2:18_1"><i><font color="Green">E19</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_1"><b>suppose </b></a><a NAME="E1:18_1_2_1"/><i><font color="Green">E21</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font>, <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:18_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><b>hence </b><a NAME="E3:18_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="jordan_a.html#E2:18"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan_a.html#E1:18_1_2_1"><i><font color="Green">E21</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_2"><b>suppose </b></a><a NAME="E1:18_1_2_2"/><i><font color="Green">E21</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>, <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:18_1_2_2"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan_a.html#E1:18_1_2_2"><i><font color="Green">E21</font></i></a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:18_1_2_2_1_1"/>
( <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> or <font color="Maroon">c<sub>4</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:18_1_2_2_1_1_1"/>
<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>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:18_1_2_2_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="jordan_a.html#E2:18"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan_a.html#E3:18_1"><i><font color="Green">E20</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="18_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:18_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>4</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/><div class="add"><b>hence </b><a NAME="E2:18_1_2_2_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="jordan_a.html#E2:18"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan_a.html#E2:18_1_2_2"><i><font color="Green">E22</font></i></a>, <a class="ref" href="jordan6.html#D10">JORDAN6:def 10</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:18_1_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <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> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> = <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> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<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> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<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> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>8</sub></font> = <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> : ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span> ;<br/>
<a NAME="E6:18"/><i><font color="Green">E19</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<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> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<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> : <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="fraenkel.html#S3">FRAENKEL:sch 3</a>(<a class="txt" href="jordan_a.html#E5:18"><i><font color="Green">E18</font></i></a>);<br/></i>
<a NAME="E7:18"/><i><font color="Green">E20</font></i>: 
 <a href="jordan7.html#K1">Segment</a> <font color="Maroon">c<sub>2</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> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<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> : <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan7.html#D1">JORDAN7:def 1</a></i>;<br/>
<a NAME="E8:18"/><i><font color="Green">E21</font></i>: 
 <a href="jordan6.html#K3">L_Segment</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<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>,<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> <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> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>by </b><i><a class="ref" href="jordan6.html#D3">JORDAN6:def 3</a></i>;<br/>
<a NAME="E9:18"/>
<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> : ( <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] or <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span>  <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> : <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span>  <a href="xboole_0.html#K2">\/</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> : <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] </span>}</span> 
 
<b>from </b><i><a class="ref" href="topreal1.html#S1">TOPREAL1:sch 1</a>();<br/></i>
<b>hence </b><a NAME="E10:18"/>
 <a href="jordan7.html#K1">Segment</a> <font color="Maroon">c<sub>2</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> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="jordan6.html#K4">R_Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<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 class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K3">L_Segment</a> <span class="p2">(<span class="default"><a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<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 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>)</span>
 <b>by </b><i><a class="txt" href="jordan_a.html#E6:18"><i><font color="Green">E19</font></i></a>, <a class="txt" href="jordan_a.html#E7:18"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan_a.html#E8:18"><i><font color="Green">E21</font></i></a>, <a class="ref" href="jordan6.html#D4">JORDAN6:def 4</a></i>;<br/>


</div>
