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<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="compts_1.html#NV6" title="COMPTS_1:NV.6">Hausdorff</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  <a href="yellow_8.html#V2" title="YELLOW_8:attr.2">irreducible</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>assume </b><a NAME="E1:28"/>
not <font color="Maroon" title="c2">S</font> is <a href="realset1.html#V1" title="REALSET1:attr.1">trivial</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">S</font><b> such that </b><br/><a NAME="E3:28"/><i><font color="Green" title="E21">A1</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="ref" href="yellow_8.html#D1" target="_self" title="YELLOW_8:def.1">Def1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">x</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">y</font> = <font color="Maroon" title="c4">y</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>consider </b><font color="Maroon" title="c7">W</font>, <font color="Maroon" title="c8">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font><b> such that </b><br/><a NAME="E6:28"/><i><font color="Green" title="E22">A2</font></i>: 
( <font color="Maroon" title="c7">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c8">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>and </b><br/><a NAME="E7:28"/><i><font color="Green" title="E23">A3</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">W</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">V</font> )
 <b>and </b><br/><a NAME="E8:28"/><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c7">W</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c8">V</font>
 <b>by </b><i><a class="txt" href="yellow_8.html#E3:28"><i><font color="Green" title="E21">A1</font></i></a>, <a class="ref" href="compts_1.html#D4" title="COMPTS_1:def.4">COMPTS_1:def 4</a></i>;<br/>
<a NAME="E9:28"/><i><font color="Green" title="E25">A5</font></i>: 
<font color="Maroon" title="c2">S</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="ref" href="yellow_8.html#D4" target="_self" title="YELLOW_8:def.4">Def4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c9">S1</font> = <font color="Maroon" title="c2">S</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c7">W</font>;<br/>
<b>set </b><font color="Maroon" title="c10">S2</font> = <font color="Maroon" title="c2">S</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">V</font>;<br/>
<a NAME="E10:28"/><i><font color="Green" title="E26">A6</font></i>: 
( <font color="Maroon" title="c2">S</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c7">W</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp; <font color="Maroon" title="c2">S</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">V</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 
<b>by </b><i><a class="txt" href="yellow_8.html#E6:28"><i><font color="Green" title="E22">A2</font></i></a>, <a class="txt" href="yellow_8.html#E9:28"><i><font color="Green" title="E25">A5</font></i></a>, <a class="ref" href="yellow_8.html#T29" target="_self" title="YELLOW_8:th.29">Th29</a></i>;<br/>
<a NAME="E11:28"/><i><font color="Green" title="E27">A7</font></i>: 
<font color="Maroon" title="c7">W</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c8">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="yellow_8.html#E8:28"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<i><font color="Green" title="E28">A8</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c7">W</font></span>)</span> <a href="finsub_1.html#K1" title="FINSUB_1:func.1">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">V</font></span>)</span> = 
<font color="Maroon" title="c2">S</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">W</font> <a href="finsub_1.html#K3" title="FINSUB_1:func.3">/\</a> <font color="Maroon" title="c8">V</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T54" title="XBOOLE_1:th.54">XBOOLE_1:54</a></i>
<br/>.= 
<font color="Maroon" title="c2">S</font>
<b>by </b><i><a class="txt" href="yellow_8.html#E11:28"><i><font color="Green" title="E27">A7</font></i></a></i>
;<br/>
<a NAME="E13:28"/>
( <font color="Maroon" title="c2">S</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c7">W</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Maroon" title="c2">S</font> <a href="finsub_1.html#K2" title="FINSUB_1:func.2">\</a> <font color="Maroon" title="c8">V</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="txt" href="yellow_8.html#E7:28"><i><font color="Green" title="E23">A3</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E14:28"/>
contradiction
 <b>by </b><i><a class="txt" href="yellow_8.html#E10:28"><i><font color="Green" title="E26">A6</font></i></a>, <a class="txt" href="yellow_8.html#E12:28"><i><font color="Green" title="E28">A8</font></i></a>, <a class="ref" href="yellow_8.html#D4" target="_self" title="YELLOW_8:def.4">Def4</a></i>;<br/>


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