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

<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>;<br/>

<a NAME="E1:27"/>
 <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="jordan6.html#T76" title="JORDAN6:th.76">JORDAN6:76</a></i>;<br/>
<a NAME="E2:27"/><b>then </b><i><font color="Green" title="E19">A1</font></i>: 
 <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T130" title="PSCOMP_1:th.130">PSCOMP_1:130</a></i>;<br/>
<a NAME="E3:27"/>
 <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1"/><i><font color="Green" title="E20">A2</font></i>: 
 <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font>
 ;<br/>

<a NAME="E2:27_1"/><i><font color="Green" title="E21">A3</font></i>: 
<span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="euclid.html#T56" title="EUCLID:th.56">EUCLID:56</a></i>;<br/>
<a NAME="E3:27_1"/><i><font color="Green" title="E22">A4</font></i>: 
 <a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_1_1"/>
 <a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c2">p</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:27_1_1"/><i><font color="Green" title="E23">A5</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>
 <b>and </b><br/><a NAME="E4:27_1_1"/><i><font color="Green" title="E24">A6</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c3">p</font> = <font color="Maroon" title="c2">p</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan21.html#E3:27_1_1"><i><font color="Green" title="E23">A5</font></i></a></i>;<br/>
<a NAME="E6:27_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan21.html#E4:27_1_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="pscomp_1.html#T71" title="PSCOMP_1:th.71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E7:27_1_1"/><b>then </b>
 <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c3">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 
<b>by </b><i><a class="txt" href="jordan21.html#E1:27_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E8:27_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan21.html#E2:27_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="txt" href="jordan21.html#E3:27_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="topreal1.html#D13" title="TOPREAL1:def.13">TOPREAL1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:27_1"/><i><font color="Green" title="E23">A7</font></i>: 
 <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K7" title="TOPREAL1:func.7">east_halfline</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<b>by </b><i><a class="ref" href="topreal1.html#T45" title="TOPREAL1:th.45">TOPREAL1:45</a></i>;<br/>
<a NAME="E5:27_1"/>
 <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="jordan7.html#T1" title="JORDAN7:th.1">JORDAN7:1</a></i>;<br/>
<b>hence </b><a NAME="E6:27_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan21.html#E3:27_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="txt" href="jordan21.html#E4:27_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:27"/>
 <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <span class="p1">(<span class="default"><a href="jordan6.html#K9" title="JORDAN6:func.9">Lower_Arc</a> <font color="Maroon" title="c1">C</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan21.html#E2:27"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/>


</div>
