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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="topreal2.html#NM1">Simple_closed_curve</a>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:21"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:21"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E3:21"/>
 <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R2">meets</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="jordan11.html#T12" target="_self">Th12</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:21"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>3</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>and </b><br/><a NAME="E6:21"/><i><font color="Green">E20</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <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>by </b><i><a class="txt" href="jordan11.html#E5:21"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E8:21"/>
 <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T28">JORDAN2C:28</a></i>;<br/>
<a NAME="E9:21"/><b>then </b><i><font color="Green">E21</font></i>: 
not <font color="Maroon">c<sub>4</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>by </b><i><a class="txt" href="jordan11.html#E5:21"><i><font color="Green">E19</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E10:21"/><b>then </b><i><font color="Green">E22</font></i>: 
 <a href="jordan1k.html#K5">dist</a> <font color="Maroon">c<sub>4</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> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan1k.html#K5">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E2:21"><i><font color="Green">E18</font></i></a>, <a class="ref" href="jordan1k.html#T51">JORDAN1K:51</a></i>;<br/>
<a NAME="E11:21"/>
 <a href="jordan2c.html#K2">UBD</a> <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#R1">misses</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#T28">JORDAN2C:28</a></i>;<br/>
<a NAME="E12:21"/><b>then </b><i><font color="Green">E23</font></i>: 
not <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E6:21"><i><font color="Green">E20</font></i></a>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<a NAME="E13:21"/><b>then </b><i><font color="Green">E24</font></i>: 
 <a href="jordan1k.html#K5">dist</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan1k.html#K5">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E1:21"><i><font color="Green">E17</font></i></a>, <a class="ref" href="jordan1k.html#T51">JORDAN1K:51</a></i>;<br/>
<a NAME="E14:21"/>
 <a href="jordan1k.html#K5">dist</a> <font color="Maroon">c<sub>4</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>4</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="txt" href="jordan11.html#E9:21"><i><font color="Green">E21</font></i></a>, <a class="ref" href="jordan11.html#T14" target="_self">Th14</a></i>;<br/>
<a NAME="E15:21"/><b>then </b>
 <a href="jordan1k.html#K5">dist</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan1k.html#K5">dist</a> <font color="Maroon">c<sub>4</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="txt" href="jordan11.html#E13:21"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:21"/><b>then </b>
 <a href="jordan1k.html#K5">dist</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="jordan2c.html#K1">BDD</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a>  <a href="jordan1k.html#K5">dist</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan11.html#E10:21"><i><font color="Green">E22</font></i></a>, <a class="ref" href="xxreal_0.html#T2">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">E23</font></i></a>, <a class="ref" href="jordan11.html#T14" target="_self">Th14</a></i>;<br/>


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