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<b>let </b><font color="Maroon">P</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="compts_1.html#V6">compact</a> <a href="struct_0.html#NM2">Subset</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="E1:18"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">P</font> <a href="topreal2.html#NR1" target="_self">is_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">E50</font></i>: 
 <a href="pscomp_1.html#K34">S-min</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K30">N-max</a> <font color="Maroon">P</font>
 ;<br/><a NAME="E2:18_1"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K10">inf</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K26">S-most</a> <font color="Maroon">P</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">P</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K34">S-min</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D49">PSCOMP_1:def 49</a></i>;<br/><a NAME="E3:18_1"/>
<span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><span class="p2">(<span class="default"><a href="pscomp_1.html#K11">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K13">proj1</a>  <a href="pscomp_1.html#K12">||</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K24">N-most</a> <font color="Maroon">P</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">P</font></span>)</span></span><a href="euclid.html#K23">]|</a></span> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K30">N-max</a> <font color="Maroon">P</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D45">PSCOMP_1:def 45</a></i>;<br/><a NAME="E4:18_1"/><b>then </b>
 <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">P</font>
 <b>by </b><i>, , <a class="ref" href="sppol_2.html#T1">SPPOL_2:1</a></i>;<br/><b>hence </b><a NAME="E5:18_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
 <a href="pscomp_1.html#K34">S-min</a> <font color="Maroon">P</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="pscomp_1.html#K30">N-max</a> <font color="Maroon">P</font>
 ;<br/>


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