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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#V6" title="COMPTS_1:attr.6">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> <a href="topreal2.html#NR1" title="TOPREAL2:NR.1">is_simple_closed_curve</a>  implies  <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-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#K28" title="PSCOMP_1:func.28">N-max</a> <font color="Maroon" title="c1">P</font> )</span><br/>

<b>assume </b><a NAME="E1:18"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c1">P</font> <a href="topreal2.html#NR1" title="TOPREAL2:NR.1">is_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#K32" title="PSCOMP_1:func.32">S-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#K28" title="PSCOMP_1:func.28">N-max</a> <font color="Maroon" title="c1">P</font></span><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="E17">A2</font></i>: 
 <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-min</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K28" title="PSCOMP_1:func.28">N-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="E2:18_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#K8" title="PSCOMP_1:func.8">inf</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">proj1</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K24" title="PSCOMP_1:func.24">S-most</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c1">P</font></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#K32" title="PSCOMP_1:func.32">S-min</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D49" title="PSCOMP_1:def.49">PSCOMP_1:def 49</a></i>;<br/><a NAME="E3:18_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#K9" title="PSCOMP_1:func.9">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K11" title="PSCOMP_1:func.11">proj1</a>  <a href="pscomp_1.html#K10" title="PSCOMP_1:func.10">||</a> <span class="p4">(<span class="default"><a href="pscomp_1.html#K22" title="PSCOMP_1:func.22">N-most</a> <font color="Maroon" title="c1">P</font></span>)</span></span>)</span></span>)</span>,<span class="p2">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c1">P</font></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#K28" title="PSCOMP_1:func.28">N-max</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#D45" title="PSCOMP_1:def.45">PSCOMP_1:def 45</a></i>;<br/><a NAME="E4:18_1"/><b>then </b>
 <a href="pscomp_1.html#K16" title="PSCOMP_1:func.16">S-bound</a> <font color="Maroon" title="c1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">N-bound</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="txt" href="topreal5.html#E1:18_1"><i><font color="Green" title="E17">A2</font></i></a>, <a class="txt" href="topreal5.html#E2:18_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="E5:18_1"/>
contradiction
 <b>by </b><i><a class="txt" href="topreal5.html#E1:18"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="topreal5.html#T22" target="_self" title="TOPREAL5:th.22">Th22</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="E3:18"/>
 <a href="pscomp_1.html#K32" title="PSCOMP_1:func.32">S-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#K28" title="PSCOMP_1:func.28">N-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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