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

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_1"/><i><font color="Green" title="E2">A1</font></i>: 
 <a href="pscomp_1.html#K26" title="PSCOMP_1:func.26">W-max</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 ;<br/><a NAME="E2:5_1"/><i><font color="Green" title="E3">A2</font></i>: 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">W-most</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K26" title="PSCOMP_1:func.26">W-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D43" title="PSCOMP_1:def.43">PSCOMP_1:def 43</a></i>;<br/><a NAME="E3:5_1"/>
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">E-bound</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K9" title="PSCOMP_1:func.9">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K12" title="PSCOMP_1:func.12">proj2</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">E-most</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D46" title="PSCOMP_1:def.46">PSCOMP_1:def 46</a></i>;<br/><a NAME="E4:5_1"/><b>then </b>
 <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">W-bound</a> <font color="Maroon" title="c1">P</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> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan1b.html#E1:5_1"><i><font color="Green" title="E2">A1</font></i></a>, <a class="txt" href="jordan1b.html#E2:5_1"><i><font color="Green" title="E3">A2</font></i></a>, <a class="ref" href="sppol_2.html#T1" title="SPPOL_2:th.1">SPPOL_2:1</a></i>;<br/><b>hence </b><a NAME="E5:5_1"/>
contradiction
 <b>by </b><i><a class="ref" href="topreal5.html#T23" title="TOPREAL5:th.23">TOPREAL5:23</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:5"/>
 <a href="pscomp_1.html#K26" title="PSCOMP_1:func.26">W-max</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="pscomp_1.html#K29" title="PSCOMP_1:func.29">E-max</a> <font color="Maroon" title="c1">P</font>
 ;<br/>


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