<?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>, <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><b>that </b><br/><a NAME="E1:8"/><i><font color="Green">E5</font></i>: 
<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>
 <b>and </b><br/><a NAME="E2:8"/><i><font color="Green">E6</font></i>: 
 <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>
 ;<br/>

<a NAME="E3:8"/><i><font color="Green">E7</font></i>: 
 <a href="jordan6.html#K8">Upper_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#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>
 
<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">E8</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#D9">JORDAN6:def 9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_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"/>
( ( <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> &amp; <font color="Maroon">c<sub>3</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; not <font color="Maroon">c<sub>3</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>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> &amp; <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>1</sub></font> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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 ( <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>1</sub></font> &amp; <font color="Maroon">c<sub>3</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; not <font color="Maroon">c<sub>3</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> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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="jordan17.html#E2:8"><i><font color="Green">E6</font></i></a>, <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"><b>suppose </b></a><b>that </b><a NAME="E1:8_1_1"/><i><font color="Green">E9</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>and </b><br/><a NAME="E2:8_1_1"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</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>and </b><br/><a NAME="E3:8_1_1"/><i><font color="Green">E11</font></i>: 
not <font color="Maroon">c<sub>3</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"><a NAME="E4:8_1_1"/><i><font color="Green">E12</font></i>: 
 <a href="pscomp_1.html#K31">E-max</a> <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>1</sub></font>
 <b>by </b><i><a class="ref" href="jordan7.html#T1">JORDAN7:1</a></i>;<br/><a NAME="E5:8_1_1"/><i><font color="Green">E13</font></i>: 
 <a href="pscomp_1.html#K31">E-max</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="E6:8_1_1"/><i><font color="Green">E14</font></i>: 
 <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</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>
 <b>by </b><i><a class="ref" href="topreal5.html#T25">TOPREAL5:25</a></i>;<br/><b>thus </b><a NAME="E7:8_1_1"/>
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#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>2</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>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_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_1_1_1_1"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font> ) or <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>1</sub></font> or <font color="Maroon">c<sub>3</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> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:8_1_1_1_1_1"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E2:8_1_1_1_1_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>take </b><font color="Maroon">c<sub>4</sub></font> =  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">c<sub>1</sub></font>;<br/><b>thus </b><a NAME="E3:8_1_1_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="jordan17.html#E1:8_1_1_1_1_1"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan17.html#E2:8_1_1_1_1_1"><i><font color="Green">E16</font></i></a></i>;<br/><b>thus </b><a NAME="E4:8_1_1_1_1_1"/>
(  <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;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan17.html#E1:8_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan17.html#E2:8_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan17.html#E3:8_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan17.html#E4:8_1_1"><i><font color="Green">E12</font></i></a>, <a class="txt" href="jordan17.html#E5:8_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan17.html#E6:8_1_1"><i><font color="Green">E14</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="8_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_2"/><i><font color="Green">E15</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>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:8_1_1_1_1_2"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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="jordan17.html#E4:8"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan17.html#E2:8_1_1"><i><font color="Green">E10</font></i></a>, <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>4</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="E4:8_1_1_1_1_2"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E5:8_1_1_1_1_2"/><i><font color="Green">E17</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#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>and </b><br/><a NAME="E6:8_1_1_1_1_2"/><i><font color="Green">E18</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</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="jordan17.html#E1:8"><i><font color="Green">E5</font></i></a>, <a class="txt" href="jordan17.html#E4:8"><i><font color="Green">E8</font></i></a>, <a class="ref" href="jordan17.html#T6" target="_self">Th6</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/><b>thus </b><a NAME="E7:8_1_1_1_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan17.html#E4:8_1_1_1_1_2"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E8:8_1_1_1_1_2"/><i><font color="Green">E19</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="jordan17.html#E5:8_1_1_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><a NAME="E9:8_1_1_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>
 <b>by </b><i><a class="txt" href="jordan17.html#E4:8"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan17.html#E3:8_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan17.html#E6:8_1_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="jordan6.html#T70">JORDAN6:70</a></i>;<br/><b>hence </b><a NAME="E10:8_1_1_1_1_2"/>
(  <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;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan17.html#E2:8_1_1"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan17.html#E3:8_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan17.html#E5:8_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan17.html#E1:8_1_1_1_1_2"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan17.html#E5:8_1_1_1_1_2"><i><font color="Green">E17</font></i></a>, <a class="txt" href="jordan17.html#E6:8_1_1_1_1_2"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan17.html#E8:8_1_1_1_1_2"><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="8_1_1_1_1_3"><b>suppose </b></a><a NAME="E1:8_1_1_1_1_3"/><i><font color="Green">E15</font></i>: 
<font color="Maroon">c<sub>3</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"><a NAME="E2:8_1_1_1_1_3"/><b>then </b>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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>
 <b>by </b><i><a class="txt" href="jordan17.html#E3:8"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan17.html#E1:8_1_1"><i><font color="Green">E9</font></i></a>, <a class="ref" href="jordan5c.html#T10">JORDAN5C:10</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>4</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="E4:8_1_1_1_1_3"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E5:8_1_1_1_1_3"/><i><font color="Green">E17</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>
 <b>and </b><br/><a NAME="E6:8_1_1_1_1_3"/>
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</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>
 <b>by </b><i><a class="txt" href="jordan17.html#E1:8"><i><font color="Green">E5</font></i></a>, <a class="txt" href="jordan17.html#E3:8"><i><font color="Green">E7</font></i></a>, <a class="ref" href="jordan17.html#T6" target="_self">Th6</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/><b>thus </b><a NAME="E7:8_1_1_1_1_3"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan17.html#E4:8_1_1_1_1_3"><i><font color="Green">E16</font></i></a></i>;<br/><a NAME="E8:8_1_1_1_1_3"/>
<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="txt" href="jordan17.html#E5:8_1_1_1_1_3"><i><font color="Green">E17</font></i></a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><b>hence </b><a NAME="E9:8_1_1_1_1_3"/>
(  <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;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan17.html#E1:8_1_1"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan17.html#E3:8_1_1"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan17.html#E5:8_1_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="jordan17.html#E1:8_1_1_1_1_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan17.html#E5:8_1_1_1_1_3"><i><font color="Green">E17</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></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:8_1_2"/><i><font color="Green">E9</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>and </b><br/><a NAME="E2:8_1_2"/><i><font color="Green">E10</font></i>: 
<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>1</sub></font>
 <b>and </b><br/><a NAME="E3:8_1_2"/><i><font color="Green">E11</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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"><b>consider </b><font color="Maroon">c<sub>4</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="E5:8_1_2"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E6:8_1_2"/><i><font color="Green">E13</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> &amp;  <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</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> )
 <b>by </b><i><a class="txt" href="jordan17.html#E1:8"><i><font color="Green">E5</font></i></a>, <a class="txt" href="jordan17.html#E3:8"><i><font color="Green">E7</font></i></a>, <a class="txt" href="jordan17.html#E3:8_1_2"><i><font color="Green">E11</font></i></a>, <a class="ref" href="jordan17.html#T6" target="_self">Th6</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/><b>thus </b><a NAME="E7:8_1_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan17.html#E5:8_1_2"><i><font color="Green">E12</font></i></a></i>;<br/><a NAME="E8:8_1_2"/>
<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="txt" href="jordan17.html#E6:8_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><b>hence </b><a NAME="E9:8_1_2"/>
(  <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;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan17.html#E1:8_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan17.html#E2:8_1_2"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan17.html#E6:8_1_2"><i><font color="Green">E13</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="8_1_3"><b>suppose </b></a><b>that </b><a NAME="E1:8_1_3"/><i><font color="Green">E9</font></i>: 
<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>1</sub></font>
 <b>and </b><br/><a NAME="E2:8_1_3"/><i><font color="Green">E10</font></i>: 
<font color="Maroon">c<sub>3</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>and </b><br/><a NAME="E3:8_1_3"/><i><font color="Green">E11</font></i>: 
not <font color="Maroon">c<sub>3</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>and </b><br/><a NAME="E4:8_1_3"/><i><font color="Green">E12</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</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"><b>consider </b><font color="Maroon">c<sub>4</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="E6:8_1_3"/><i><font color="Green">E13</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E7:8_1_3"/><i><font color="Green">E14</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#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>and </b><br/><a NAME="E8:8_1_3"/><i><font color="Green">E15</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</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="jordan17.html#E1:8"><i><font color="Green">E5</font></i></a>, <a class="txt" href="jordan17.html#E4:8"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan17.html#E4:8_1_3"><i><font color="Green">E12</font></i></a>, <a class="ref" href="jordan17.html#T6" target="_self">Th6</a></i>;<br/><b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/><b>thus </b><a NAME="E9:8_1_3"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> )
 <b>by </b><i><a class="txt" href="jordan17.html#E6:8_1_3"><i><font color="Green">E13</font></i></a></i>;<br/><a NAME="E10:8_1_3"/><i><font color="Green">E16</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>
 <b>by </b><i><a class="txt" href="jordan17.html#E4:8"><i><font color="Green">E8</font></i></a>, <a class="txt" href="jordan17.html#E3:8_1_3"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan17.html#E8:8_1_3"><i><font color="Green">E15</font></i></a>, <a class="ref" href="jordan6.html#T70">JORDAN6:70</a></i>;<br/><a NAME="E11:8_1_3"/>
<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="jordan17.html#E7:8_1_3"><i><font color="Green">E14</font></i></a>, <a class="ref" href="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/><b>hence </b><a NAME="E12:8_1_3"/>
(  <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;  <a href="jordan6.html#R1">LE</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan17.html#E1:8_1_3"><i><font color="Green">E9</font></i></a>, <a class="txt" href="jordan17.html#E2:8_1_3"><i><font color="Green">E10</font></i></a>, <a class="txt" href="jordan17.html#E3:8_1_3"><i><font color="Green">E11</font></i></a>, <a class="txt" href="jordan17.html#E7:8_1_3"><i><font color="Green">E14</font></i></a>, <a class="txt" href="jordan17.html#E8:8_1_3"><i><font color="Green">E15</font></i></a>, <a class="txt" href="jordan17.html#E10:8_1_3"><i><font color="Green">E16</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>
