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<span class="kw">let </span><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#NV4" title="COMPTS_1:NV.4">Hausdorff</a>  <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">S</font> being  <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">irreducible</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds  <font color="Olive" title="b1">S</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="yellow_8.html#V3" title="YELLOW_8:attr.3">irreducible</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">S</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> </span><br/>



<span class="kw">assume </span><a NAME="E1:39"/>
not <font color="Maroon" title="c2">S</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:39"/><span class="lab"><font color="Green" title="E21">A1</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_8.html#D1" target="_self" title="YELLOW_8:def.1">Def1</a></span>;<br/>
<span class="kw">reconsider </span><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/>
<span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E6:39"/><span class="lab"><font color="Green" title="E22">A2</font></span>: 
( <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>  )
 <span class="kw">and </span><br/><a NAME="E7:39"/><span class="lab"><font color="Green" title="E23">A3</font></span>: 
( <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> )
 <span class="kw">and </span><br/><a NAME="E8:39"/><span class="lab"><font color="Green" title="E24">A4</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:39"><span class="lab"><font color="Green" title="E21">A1</font></span></a>, <a class="ref" href="pre_topc.html#D16" title="PRE_TOPC:def.16">PRE_TOPC:def 16</a></span>;<br/>
<span class="kw">set </span><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/>
<span class="kw">set </span><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="E9:39"/><span class="lab"><font color="Green" title="E25">A5</font></span>: 
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E23">A3</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/>
<a NAME="E10:39"/>
<font color="Maroon" title="c2">S</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_8.html#D4" target="_self" title="YELLOW_8:def.4">Def4</a></span>;<br/>
<a NAME="E11:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A6</font></span>: 
( <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>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:39"><span class="lab"><font color="Green" title="E22">A2</font></span></a>, <a class="ref" href="yellow_8.html#T29" target="_self" title="YELLOW_8:th.29">Th29</a></span>;<br/>
<a NAME="E12:39"/><span class="lab"><font color="Green" title="E27">A7</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:39"><span class="lab"><font color="Green" title="E24">A4</font></span></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></span>;<br/>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T54" title="XBOOLE_1:th.54">XBOOLE_1:54</a></span>
<br/>.= 
<font color="Maroon" title="c2">S</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:39"><span class="lab"><font color="Green" title="E27">A7</font></span></a></span>
;<br/>
<span class="kw">hence </span><a NAME="E14:39"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:39"><span class="lab"><font color="Green" title="E26">A6</font></span></a>, <a class="txt" href="#E9:39"><span class="lab"><font color="Green" title="E25">A5</font></span></a>, <a class="ref" href="yellow_8.html#D4" target="_self" title="YELLOW_8:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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