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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="18_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:18_1"/><i><font color="Green" title="E10">A1</font></i>: 
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">E-min</a> <font color="Maroon" title="c1">P</font>
 ;<br/><a NAME="E2:18_1"/>
 <a href="pscomp_1.html#K15" title="PSCOMP_1:func.15">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#K17" title="PSCOMP_1:func.17">E-bound</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="jordan1e.html#E1:18_1"><i><font color="Green" title="E10">A1</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="E3:18_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="E2:18"/>
 <a href="pscomp_1.html#K27" title="PSCOMP_1:func.27">W-min</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#K32" title="PSCOMP_1:func.32">E-min</a> <font color="Maroon" title="c1">P</font>
 ;<br/>


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