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<b>let </b><font color="Maroon">P</font> be   <a href="topreal2.html#NM1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon">a</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><a NAME="E1:5"/><i><font color="Green">E32</font></i>: 
 <a href="jordan6.html#R1">LE</a> <font color="Maroon">a</font>, <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>,<font color="Maroon">P</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1"/>
( ( <font color="Maroon">a</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#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; not  <a href="pscomp_1.html#K27">W-min</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">a</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#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not  <a href="pscomp_1.html#K27">W-min</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">a</font>, <a href="pscomp_1.html#K27">W-min</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> ) or ( <font color="Maroon">a</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#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="jordan5c.html#R1">LE</a> <font color="Maroon">a</font>, <a href="pscomp_1.html#K27">W-min</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> ) )
 </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="5_1_1"><b>suppose </b></a><a NAME="E1:5_1_1"/>
( ( <font color="Maroon">a</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#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; not  <a href="pscomp_1.html#K27">W-min</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">a</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#K9">Lower_Arc</a> <font color="Maroon">P</font> &amp; not  <a href="pscomp_1.html#K27">W-min</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">a</font>, <a href="pscomp_1.html#K27">W-min</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"><b>hence </b><a NAME="E2:5_1_1"/>
<font color="Maroon">a</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="5_1_2"><b>suppose </b></a><b>that </b><a NAME="E1:5_1_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E2:5_1_2"/>
 <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>
 <b>and </b><br/><a NAME="E3:5_1_2"/><i><font color="Green">E34</font></i>: 
 <a href="jordan5c.html#R1">LE</a> <font color="Maroon">a</font>, <a href="pscomp_1.html#K27">W-min</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"><a NAME="E4:5_1_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#D8">JORDAN6:def 8</a></i>;<br/><b>then consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jordan6.html#K8">Upper_Arc</a> <font color="Maroon">P</font></span>)</span></span>)</span>, <font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:5_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>and </b><br/><a NAME="E7:5_1_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K27">W-min</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E8:5_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K31">E-max</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E9:5_1_2"/><i><font color="Green">E38</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E10:5_1_2"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 <b>and </b><br/><a NAME="E11:5_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><b>thus </b><a NAME="E12:5_1_2"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K9">Lower_Arc</a> <font color="Maroon">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:5_1_2_1_1"/>
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:5_1_2_1_1_1"/>
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>hence </b><a NAME="E2:5_1_2_1_1_1"/>
<font color="Maroon">a</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="jordan7.html#T1">JORDAN7:1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="5_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:5_1_2_1_1_2"/>
<font color="Maroon">r</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:5_1_2_1_1_2"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:5_1_2_1_1_2"/>
<font color="Maroon">a</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="jordan5c.html#D3">JORDAN5C:def 3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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

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