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<div class="add">

<b>let </b><font color="Maroon">C</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="E1:16"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">C</font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a> 
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E50</font></i>: 
 <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font>
 ;<br/><a NAME="E2:16_1"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font> holds <br/><font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</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="E1:16_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">C</font>
 ;<br/>

<a NAME="E2:16_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
(  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font> )
 
<b>by </b><i><a class="ref" href="pscomp_1.html#T71">PSCOMP_1:71</a></i>;<br/>
<a NAME="E3:16_1_1"/><b>then </b>
 <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font>
 
<b>by </b><i><a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E4:16_1_1"/><b>then </b>
 <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font> <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>
<b>hence </b><a NAME="E5:16_1_1"/>
<font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:16_1"/>
contradiction
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:16"/>
 <a href="pscomp_1.html#K16">N-bound</a> <font color="Maroon">C</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="pscomp_1.html#K18">S-bound</a> <font color="Maroon">C</font>
 ;<br/>


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