<?xml version="1.0"?>
<div class="add">

<b>set </b><font color="Maroon">Cb</font> =  <a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1_1"/>
( <font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 or <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 or <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">r</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0
 ;<br/><div class="add"><b>set </b><font color="Maroon">A</font> = <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p1">(<span class="default"><a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font></span>)</span>;<br/><a NAME="E2:41_1_1_1_1"/><i><font color="Green">E30</font></i>: 
 <a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_1_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:41_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font>
 ;<br/><a NAME="E2:41_1_1_1_1_1_1"/><b>then </b>
 ex <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 <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">p</font> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 ;<br/><b>hence </b><a NAME="E3:41_1_1_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="jgraph_6.html#E10"><i><font color="Green">Lemma40</font></i></a>, <a class="ref" href="toprns_1.html#T26">TOPRNS_1:26</a></i>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:41_1_1_1_1_1"/>
 <a href="xboole_0.html#K1">{}</a>  <a href="tarski.html#R1">c=</a>  <a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E3:41_1_1_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E11"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/><a NAME="E4:41_1_1_1_1"/>
for <font color="Olive">P</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font></span>)</span></span>)</span>  st <font color="Olive">P</font> <a href="hidden.html#R1">=</a>  <a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font> holds <br/><font color="Olive">P</font> is <a href="compts_1.html#V6">compact</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:41_1_1_1_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R1">=</a>  <a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font>
 ;<br/>

<a NAME="E2:41_1_1_1_1_2"/>
 <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <span class="p2">(<span class="default"><a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font></span>)</span></span>)</span> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i><a class="ref" href="compts_1.html#T9">COMPTS_1:9</a></i>;<br/>
<b>hence </b><a NAME="E3:41_1_1_1_1_2"/>
<font color="Maroon">P</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E11"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="jgraph_6.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E5:41_1_1_1_1"/>
 <a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E12"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="compts_1.html#T11">COMPTS_1:11</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_1_2"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:41_1_1_1_2"/><b>then </b>
 <a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font> <a href="topreal2.html#NR1" target="_self">is_simple_closed_curve</a> 
 <b>by </b><i><a class="ref" href="jgraph_6.html#T7" target="_self">Th7</a></i>;<br/><b>hence </b><a NAME="E3:41_1_1_1_2"/>
 <a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_3"><b>suppose </b></a><a NAME="E1:41_1_1_1_3"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:41_1_1_1_3"/>
 <a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span></span><a href="tarski.html#K1">}</a></span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_3_1"><b>proof </b></a><div class="add">

<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a>
<div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:41_1_1_1_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font>
 ;<br/><b>then consider </b><font color="Maroon">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><b> such that </b><br/><a NAME="E3:41_1_1_1_3_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E4:41_1_1_1_3_1_1"/><i><font color="Green">E38</font></i>: 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><a NAME="E5:41_1_1_1_3_1_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E10"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="jgraph_6.html#E12"><i><font color="Green">Lemma42</font></i></a>, <a class="ref" href="toprns_1.html#T29">TOPRNS_1:29</a></i>;<br/><b>hence </b><a NAME="E6:41_1_1_1_3_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E11"><i><font color="Green">Lemma41</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:41_1_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span></span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E3:41_1_1_1_3_1"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E4:41_1_1_1_3_1"/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span> <a href="euclid.html#K20">-</a> <span class="p3"><a href="euclid.html#K23">|[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="euclid.html#K23">]|</a></span></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="toprns_1.html#T29">TOPRNS_1:29</a></i>;<br/>
<b>hence </b><a NAME="E5:41_1_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jgraph_6.html#E10"><i><font color="Green">Lemma40</font></i></a>, <a class="txt" href="jgraph_6.html#E11"><i><font color="Green">Lemma41</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:41_1_1_1_3"/>
 <a href="jgraph_6.html#K5" target="_self">circle</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">r</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T41">BORSUK_1:41</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
