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<b>let </b><font color="Maroon" title="c1">p</font> be   <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>;<br/><b>let </b><font color="Maroon" title="c2">C</font> be  <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</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>assume </b><a NAME="E1:21"/><i><font color="Green" title="E18">A1</font></i>: 
<font color="Maroon" title="c1">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">C</font>
 ;<br/>

<b>assume </b><a NAME="E2:21"/><i><font color="Green" title="E19">A2</font></i>: 
not  <a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">C</font>
 ;<br/>

<a NAME="E3:21"/>
( <span class="p1">(<span class="default"><a href="jordan18.html#K1" title="JORDAN18:func.1">North-Bound</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K3" title="PSCOMP_1:func.3">inf</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K6" title="TOPREAL1:func.6">north_halfline</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="jordan18.html#K2" title="JORDAN18:func.2">South-Bound</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">C</font></span>)</span> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="pscomp_1.html#K2" title="PSCOMP_1:func.2">sup</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K14" title="PSCOMP_1:func.14">proj2</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">C</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="topreal1.html#K8" title="TOPREAL1:func.8">south_halfline</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan18.html#E26"><i><font color="Green" title="E14">Lm4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:21"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan18.html#E1:21"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan18.html#E2:21"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="jordan18.html#T24" target="_self" title="JORDAN18:th.24">Th24</a></i>;<br/>


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