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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="topreal2.html#V1">being_simple_closed_curve</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>let </b><font color="Maroon">c<sub>2</sub></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/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:20_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> or  not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_1"><b>suppose </b></a><a NAME="E1:20_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:20_1_1"/><b>then </b>
 <a href="jordan1k.html#K5">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="jordan1k.html#T45">JORDAN1K:45</a></i>;<br/><b>hence </b><a NAME="E3:20_1_1"/>
( not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>1</sub></font> implies  <a href="jordan1k.html#K5">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="jordan1k.html#K5">dist</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 <b>by </b><i><a class="ref" href="jordan1k.html#T44">JORDAN1K:44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="20_1_2"><b>suppose </b></a><a NAME="E1:20_1_2"/>
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><a NAME="E2:20_1_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K3">`</a> 
 <b>by </b><i><a class="ref" href="subset_1.html#T50">SUBSET_1:50</a></i>;<br/><a NAME="E3:20_1_2"/><b>then </b><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="jordan2c.html#T31">JORDAN2C:31</a></i>;<br/><b>assume </b><b>that </b><br/><a NAME="E4:20_1_2"/><i><font color="Green">E17</font></i>: 
not <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:20_1_2"/><i><font color="Green">E18</font></i>: 
 <a href="jordan1k.html#K5">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a>  <a href="jordan1k.html#K5">dist</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E6:20_1_2"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan11.html#E3:20_1_2"><i><font color="Green">E16</font></i></a>, <a class="txt" href="jordan11.html#E4:20_1_2"><i><font color="Green">E17</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E7:20_1_2"/>
ex <font color="Olive">b<sub>1</sub></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="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="topreal6.html#K1">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan1k.html#K5">dist</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="jordan11.html#E5:20_1_2"><i><font color="Green">E18</font></i></a>, <a class="ref" href="jordan1k.html#T48">JORDAN1K:48</a></i>;<br/><b>hence </b><a NAME="E8:20_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan11.html#E6:20_1_2"><i><font color="Green">E19</font></i></a>, <a class="ref" href="jordan11.html#T13" target="_self">Th13</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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