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<b>let </b><font color="Maroon" title="c1">A</font>, <font color="Maroon" title="c2">B</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">s</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:2"/><i><font color="Green" title="E1">A1</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E2:2"/><i><font color="Green" title="E2">A2</font></i>: 
<font color="Maroon" title="c1">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan6.html#K7" title="JORDAN6:func.7">Horizontal_Line</a> <font color="Maroon" title="c3">s</font>
 <b>and </b><br/><a NAME="E3:2"/><i><font color="Green" title="E3">A3</font></i>: 
<font color="Maroon" title="c2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="jordan6.html#K7" title="JORDAN6:func.7">Horizontal_Line</a> <font color="Maroon" title="c3">s</font>
 ;<br/>

<b>assume </b><a NAME="E4:2"/>
<a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">A</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:2"/><i><font color="Green" title="E4">A4</font></i>: 
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E7:2"/><i><font color="Green" title="E5">A5</font></i>: 
<font color="Maroon" title="c4">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c2">B</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="c5">e</font> = <font color="Maroon" title="c4">e</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan15.html#E6:2"><i><font color="Green" title="E4">A4</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">c</font> being   <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> such that </b><br/><a NAME="E10:2"/><i><font color="Green" title="E6">A6</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">A</font>
 <b>and </b><br/><a NAME="E11:2"/><i><font color="Green" title="E7">A7</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">c</font>
 <b>by </b><i><a class="txt" href="jordan15.html#E6:2"><i><font color="Green" title="E4">A4</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">d</font> being   <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> such that </b><br/><a NAME="E13:2"/><i><font color="Green" title="E8">A8</font></i>: 
<font color="Maroon" title="c7">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">B</font>
 <b>and </b><br/><a NAME="E14:2"/><i><font color="Green" title="E9">A9</font></i>: 
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">d</font>
 <b>by </b><i><a class="txt" href="jordan15.html#E7:2"><i><font color="Green" title="E5">A5</font></i></a>, <a class="ref" href="funct_2.html#T116" title="FUNCT_2:th.116">FUNCT_2:116</a></i>;<br/>
<a NAME="E15:2"/>
( <font color="Maroon" title="c6">c</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> &amp; <font color="Maroon" title="c7">d</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">s</font> )
 
<b>by </b><i><a class="txt" href="jordan15.html#E2:2"><i><font color="Green" title="E2">A2</font></i></a>, <a class="txt" href="jordan15.html#E3:2"><i><font color="Green" title="E3">A3</font></i></a>, <a class="txt" href="jordan15.html#E10:2"><i><font color="Green" title="E6">A6</font></i></a>, <a class="txt" href="jordan15.html#E13:2"><i><font color="Green" title="E8">A8</font></i></a>, <a class="ref" href="jordan6.html#T35" title="JORDAN6:th.35">JORDAN6:35</a></i>;<br/>
<b>then </b><font color="Maroon" title="c6">c</font> = 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">c</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><font color="Maroon" title="c5">e</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
<b>by </b><i><a class="txt" href="jordan15.html#E11:2"><i><font color="Green" title="E7">A7</font></i></a>, <a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K23" title="EUCLID:func.23">|[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> </span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c7">d</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> </span>)</span></span><a href="euclid.html#K23" title="EUCLID:func.23">]|</a></span>
<b>by </b><i><a class="txt" href="jordan15.html#E14:2"><i><font color="Green" title="E9">A9</font></i></a>, <a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
<br/>.= 
<font color="Maroon" title="c7">d</font>
<b>by </b><i><a class="ref" href="euclid.html#T57" title="EUCLID:th.57">EUCLID:57</a></i>
;<br/>
<b>hence </b><a NAME="E17:2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan15.html#E1:2"><i><font color="Green" title="E1">A1</font></i></a>, <a class="txt" href="jordan15.html#E10:2"><i><font color="Green" title="E6">A6</font></i></a>, <a class="txt" href="jordan15.html#E13:2"><i><font color="Green" title="E8">A8</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/>


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