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<b>let </b><font color="Maroon" title="c1">C</font> be  <a href="topreal2.html#V1" title="TOPREAL2:attr.1">being_simple_closed_curve</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>;<br/><b>let </b><font color="Maroon" title="c2">D</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">D</font>
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green" title="E18">A2</font></i>: 
<font color="Maroon" title="c2">D</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 ;<br/>

<a NAME="E3:21"/>
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R2" title="SUBSET_1:pred.2">meets</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="ref" href="jordan11.html#T12" target="_self" title="JORDAN11:th.12">Th12</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:21"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E6:21"/><i><font color="Green" title="E20">A4</font></i>: 
<font color="Maroon" title="c3">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c2">D</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c3">e</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan11.html#E5:21"><i><font color="Green" title="E19">A3</font></i></a></i>;<br/>
<a NAME="E8:21"/>
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T28" title="JORDAN2C:th.28">JORDAN2C:28</a></i>;<br/>
<a NAME="E9:21"/><b>then </b><i><font color="Green" title="E21">A5</font></i>: 
not <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E5:21"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E10:21"/><b>then </b><i><font color="Green" title="E22">A6</font></i>: 
 <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c4">p</font>,<span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E2:21"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="jordan1k.html#T51" title="JORDAN1K:th.51">JORDAN1K:51</a></i>;<br/>
<a NAME="E11:21"/>
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c2">D</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T28" title="JORDAN2C:th.28">JORDAN2C:28</a></i>;<br/>
<a NAME="E12:21"/><b>then </b><i><font color="Green" title="E23">A7</font></i>: 
not <font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E6:21"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E13:21"/><b>then </b><i><font color="Green" title="E24">A8</font></i>: 
 <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c4">p</font>,<span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E1:21"><i><font color="Green" title="E17">A1</font></i></a>, <a class="ref" href="jordan1k.html#T51" title="JORDAN1K:th.51">JORDAN1K:51</a></i>;<br/>
<a NAME="E14:21"/>
 <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c1">C</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c4">p</font>,<span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan11.html#E9:21"><i><font color="Green" title="E21">A5</font></i></a>, <a class="ref" href="jordan11.html#T14" target="_self" title="JORDAN11:th.14">Th14</a></i>;<br/>
<a NAME="E15:21"/><b>then </b>
 <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c4">p</font>,<span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c4">p</font>,<span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c1">C</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan11.html#E13:21"><i><font color="Green" title="E24">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:21"/><b>then </b>
 <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c4">p</font>,<span class="p1">(<span class="default"><a href="jordan2c.html#K1" title="JORDAN2C:func.1">BDD</a> <font color="Maroon" title="c2">D</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a>  <a href="jordan1k.html#K5" title="JORDAN1K:func.5">dist</a> <font color="Maroon" title="c4">p</font>,<font color="Maroon" title="c2">D</font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E10:21"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<b>hence </b><a NAME="E17:21"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan11.html#E12:21"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="jordan11.html#T14" target="_self" title="JORDAN11:th.14">Th14</a></i>;<br/>


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