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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="compts_1.html#V3">Hausdorff</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">S</font> be  <a href="yellow_8.html#V2" target="_self">irreducible</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:28"/>
not <font color="Maroon">S</font> is <a href="realset1.html#V1">trivial</a>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font>, <font color="Maroon">y</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E3:28"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">y</font>
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">y</font> = <font color="Maroon">y</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">T</font> ;<br/>
<b>consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>, <font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E6:28"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> )
 <b>and </b><br/><a NAME="E7:28"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> )
 <b>and </b><br/><a NAME="E8:28"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 <b>by </b><i><a class="ref" href="yellow_8.html#T2" target="_self">Th2</a>, <a class="ref" href="compts_1.html#D4">COMPTS_1:def 4</a></i>;<br/>
<a NAME="E9:28"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i/>;<br/>
<b>set </b><font color="Maroon">S1</font> = <font color="Maroon">S</font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">W</font>;<br/>
<b>set </b><font color="Maroon">S2</font> = <font color="Maroon">S</font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">V</font>;<br/>
<a NAME="E10:28"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">S</font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">W</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">S</font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">V</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i>, , <a class="txt" href="yellow_8.html#E3:28"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E11:28"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">W</font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<i><font color="Green">E42</font></i>: <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">W</font></span>)</span> <a href="finsub_1.html#K1">\/</a> <span class="p1">(<span class="default"><font color="Maroon">S</font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">V</font></span>)</span> = 
<font color="Maroon">S</font> <a href="finsub_1.html#K2">\</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="finsub_1.html#K3">/\</a> <font color="Maroon">V</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T54">XBOOLE_1:54</a></i>
<br/>.= 
<font color="Maroon">S</font>
<b>by </b><i><a class="txt" href="yellow_8.html#E7:28"><i><font color="Green">E27</font></i></a></i>
;<br/>
<a NAME="E13:28"/>
( <font color="Maroon">S</font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">W</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">S</font> &amp; <font color="Maroon">S</font> <a href="finsub_1.html#K2">\</a> <font color="Maroon">V</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">S</font> )
 
<b>by </b><i><a class="ref" href="yellow_8.html#D1" target="_self">Def1</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E14:28"/>
contradiction
 <b>by </b><i>, , </i>;<br/>


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