<?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/>

<a NAME="E1:25"/>
 <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/>
<a NAME="E2:25"/><b>then </b><i><font color="Green">E18</font></i>: 
 <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K17">E-bound</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="pscomp_1.html#T130">PSCOMP_1:130</a></i>;<br/>
<a NAME="E3:25"/>
 <a href="pscomp_1.html#K17">E-bound</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> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1"/><i><font color="Green">E19</font></i>: 
 <a href="pscomp_1.html#K17">E-bound</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> <a href="xxreal_0.html#R1">&lt;</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:25_1"/><i><font color="Green">E20</font></i>: 
<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> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56">EUCLID:56</a></i>;<br/>
<a NAME="E3:25_1"/><i><font color="Green">E21</font></i>: 
 <a href="topreal1.html#K7">east_halfline</a> <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> <a href="subset_1.html#R1">misses</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:25_1_1"/>
 <a href="topreal1.html#K7">east_halfline</a> <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> <a href="subset_1.html#R2">meets</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:25_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="topreal1.html#K7">east_halfline</a> <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>
 <b>and </b><br/><a NAME="E4:25_1_1"/><i><font color="Green">E23</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="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></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> <b>by </b><i><a class="txt" href="jordan21.html#E3:25_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>
<a NAME="E6:25_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K17">E-bound</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="txt" href="jordan21.html#E4:25_1_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E7:25_1_1"/><b>then </b>
 <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="euclid.html#K21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan21.html#E1:25_1"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E8:25_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan21.html#E2:25_1"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan21.html#E3:25_1_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="topreal1.html#D13">TOPREAL1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:25_1"/><i><font color="Green">E22</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="topreal1.html#K7">east_halfline</a> <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>
 
<b>by </b><i><a class="ref" href="topreal1.html#T45">TOPREAL1:45</a></i>;<br/>
<a NAME="E5:25_1"/>
 <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/>
<b>hence </b><a NAME="E6:25_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan21.html#E3:25_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan21.html#E4:25_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:25"/>
 <a href="pscomp_1.html#K17">E-bound</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K17">E-bound</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="txt" href="jordan21.html#E2:25"><i><font color="Green">E18</font></i></a>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div>
