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<b>let </b><font color="Maroon">P</font> be   <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>assume </b><b>that </b><br/><a NAME="E1:22"/><i><font color="Green">E34</font></i>: 
not <font color="Maroon">P</font> is <a href="realset1.html#V1">trivial</a>
 <b>and </b><br/><a NAME="E2:22"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/><font color="Olive">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="euclid.html#K22">`2</a> 
 <b>and </b><br/><a NAME="E3:22"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being  <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>  st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Olive">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> holds <br/><font color="Olive">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Olive">q</font> <a href="euclid.html#K21">`1</a> 
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="sppol_1.html#D2" target="_self">SPPOL_1:def 2</a>,<a class="ref" href="sppol_1.html#D3" target="_self">SPPOL_1:def 3</a><br/>

<b>consider </b><font color="Maroon">p</font> being   <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>, <font color="Maroon">q</font> being   <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> such that </b><br/><a NAME="E5:22"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> )
 <b>and </b><br/><a NAME="E6:22"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">q</font>
 <b>by </b><i>, <a class="ref" href="realset1.html#T15">REALSET1:15</a></i>;<br/>
<a NAME="E7:22"/>
( <font color="Maroon">p</font> <a href="euclid.html#K22">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="euclid.html#K22">`2</a>  &amp; <font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="euclid.html#K21">`1</a>  )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E8:22"/>
contradiction
 <b>by </b><i>, <a class="ref" href="topreal3.html#T11">TOPREAL3:11</a></i>;<br/>


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