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<b>let </b><font color="Maroon" title="c1">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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( not <font color="Maroon" title="c1">P</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> &amp; <font color="Maroon" title="c1">P</font> is <a href="sppol_1.html#V1" title="SPPOL_1:attr.1">horizontal</a> implies  not <font color="Maroon" title="c1">P</font> is <a href="sppol_1.html#V2" title="SPPOL_1:attr.2">vertical</a> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E1:20"/><i><font color="Green" title="E12">A1</font></i>: 
not <font color="Maroon" title="c1">P</font> is <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>
 <b>and </b><br/><a NAME="E2:20"/><i><font color="Green" title="E13">A2</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> holds <br/><font color="Olive" title="b1">p</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a> 
 <b>and </b><br/><a NAME="E3:20"/><i><font color="Green" title="E14">A3</font></i>: 
for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</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>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Olive" title="b2">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> holds <br/><font color="Olive" title="b1">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="sppol_1.html#D2" target="_self" title="SPPOL_1:def.2">SPPOL_1:def 2</a>,<a class="ref" href="sppol_1.html#D3" target="_self" title="SPPOL_1:def.3">SPPOL_1:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>consider </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</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="E5:20"/><i><font color="Green" title="E15">A4</font></i>: 
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c3">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> )
 <b>and </b><br/><a NAME="E6:20"/><i><font color="Green" title="E16">A5</font></i>: 
<font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">q</font>
 <b>by </b><i><a class="txt" href="#E1:20"><i><font color="Green" title="E12">A1</font></i></a>, <a class="ref" href="realset1.html#T15" title="REALSET1:th.15">REALSET1:15</a></i>;<br/>
<a NAME="E7:20"/>
( <font color="Maroon" title="c2">p</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">q</font> <a href="euclid.html#K22" title="EUCLID:func.22">`2</a>  &amp; <font color="Maroon" title="c2">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  )
 
<b>by </b><i><a class="txt" href="#E2:20"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="#E3:20"><i><font color="Green" title="E14">A3</font></i></a>, <a class="txt" href="#E5:20"><i><font color="Green" title="E15">A4</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:20"/>
contradiction
 <b>by </b><i><a class="txt" href="#E6:20"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="topreal3.html#T11" title="TOPREAL3:th.11">TOPREAL3:11</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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