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<b>let </b><font color="Maroon" title="c1">P</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">P</font> is <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a> implies  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">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#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font> )</span><br/>

<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c1">P</font> is <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">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#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_1"/><i><font color="Green" title="E17">A2</font></i>: 
 <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">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#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:19_1"/><i><font color="Green" title="E18">A3</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#K11" title="PSCOMP_1:func.11">W-bound</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K6" title="PSCOMP_1:func.6">inf</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">proj2</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K19" title="PSCOMP_1:func.19">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#K23" title="PSCOMP_1:func.23">W-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D42" title="PSCOMP_1:def.42">PSCOMP_1:def 42</a></i>;<br/><a NAME="E3:19_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#K13" title="PSCOMP_1:func.13">E-bound</a> <font color="Maroon" title="c1">P</font></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K7" title="PSCOMP_1:func.7">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">proj2</a>  <a href="pscomp_1.html#K8" title="PSCOMP_1:func.8">|</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K21" title="PSCOMP_1:func.21">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#K27" title="PSCOMP_1:func.27">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="E4:19_1"/><b>then </b>
 <a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">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#K13" title="PSCOMP_1:func.13">E-bound</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="#E1:19_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="#E2:19_1"><i><font color="Green" title="E18">A3</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="E5:19_1"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:19"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="topreal5.html#T23" target="_self" title="TOPREAL5:th.23">Th23</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:19"/>
 <a href="pscomp_1.html#K23" title="PSCOMP_1:func.23">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#K27" title="PSCOMP_1:func.27">E-max</a> <font color="Maroon" title="c1">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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