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<b>let </b><font color="Maroon" title="c1">C</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>;<br/><b>let </b><font color="Maroon" title="c2">P</font> be   <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="c3">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:22"/><i><font color="Green" title="E16">A1</font></i>: 
<font color="Maroon" title="c3">n</font> <a href="jordan1h.html#R1" title="JORDAN1H:pred.1">is_sufficiently_large_for</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E2:22"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c2">P</font> <a href="jordan2c.html#R2" title="JORDAN2C:pred.2">is_outside_component_of</a> <font color="Maroon" title="c1">C</font>
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green" title="E18">A3</font></i>: 
<font color="Maroon" title="c2">P</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">n</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon" title="c4">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E5:22"/><i><font color="Green" title="E19">A4</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P</font> &amp; <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K5" title="TOPREAL1:func.5">L~</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">n</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan14.html#E3:22"><i><font color="Green" title="E18">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="E6:22"/>
<font color="Maroon" title="c2">P</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font>
 
<b>by </b><i><a class="txt" href="jordan14.html#E2:22"><i><font color="Green" title="E17">A2</font></i></a>, <a class="ref" href="jordan2c.html#T27" title="JORDAN2C:th.27">JORDAN2C:27</a></i>;<br/>
<a NAME="E7:22"/><b>then </b><i><font color="Green" title="E20">A5</font></i>: 
<font color="Maroon" title="c4">x</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>by </b><i><a class="txt" href="jordan14.html#E5:22"><i><font color="Green" title="E19">A4</font></i></a></i>;<br/>
<a NAME="E8:22"/>
 <a href="jordan2c.html#K2" title="JORDAN2C:func.2">UBD</a> <font color="Maroon" title="c1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="goboard9.html#K2" title="GOBOARD9:func.2">LeftComp</a> <span class="p1">(<span class="default"><a href="jordan13.html#K1" title="JORDAN13:func.1">Span</a> <font color="Maroon" title="c1">C</font>,<font color="Maroon" title="c3">n</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan14.html#E1:22"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="jordan14.html#T17" target="_self" title="JORDAN14:th.17">Th17</a></i>;<br/>
<b>hence </b><a NAME="E9:22"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan14.html#E5:22"><i><font color="Green" title="E19">A4</font></i></a>, <a class="txt" href="jordan14.html#E7:22"><i><font color="Green" title="E20">A5</font></i></a>, <a class="ref" href="gobrd14.html#T27" title="GOBRD14:th.27">GOBRD14:27</a></i>;<br/>


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