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<b>let </b><font color="Maroon" title="c1">TS</font> be   <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">PS</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green" title="E13">A1</font></i>: 
<font color="Maroon" title="c1">TS</font> <a href="compts_1.html#NR2" title="COMPTS_1:NR.2">is_T2</a> 
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green" title="E14">A2</font></i>: 
<font color="Maroon" title="c2">PS</font> is <a href="compts_1.html#V6" title="COMPTS_1:attr.6">compact</a>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:13_1"/>
( <font color="Maroon" title="c2">PS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">PS</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_1"><b>suppose </b></a><a NAME="E1:13_1_1"/>
<font color="Maroon" title="c2">PS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:13_1_1"/><b>then </b>
<font color="Maroon" title="c2">PS</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</a> <font color="Maroon" title="c1">TS</font>
 ;<br/><b>hence </b><a NAME="E3:13_1_1"/>
<font color="Maroon" title="c2">PS</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T22" title="TOPS_1:th.22">TOPS_1:22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="13_1_2"><b>suppose </b></a><a NAME="E1:13_1_2"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c2">PS</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c3">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>thus </b><a NAME="E1:13_1_2_1"/>
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PS</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  implies  ex <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font> st <br/>( <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">PS</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="13_1_2_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:13_1_2_1_1"/><i><font color="Green" title="E16">A4</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PS</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">p</font> = <font color="Maroon" title="c3">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <font color="Maroon" title="c1">TS</font> ;<br/>
<b>consider </b><font color="Maroon" title="c5">W</font>, <font color="Maroon" title="c6">V</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font><b> such that </b><br/><a NAME="E4:13_1_2_1_1"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c5">W</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E5:13_1_2_1_1"/>
<font color="Maroon" title="c6">V</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E6:13_1_2_1_1"/><i><font color="Green" title="E18">A6</font></i>: 
<font color="Maroon" title="c4">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">W</font>
 <b>and </b><br/><a NAME="E7:13_1_2_1_1"/><i><font color="Green" title="E19">A7</font></i>: 
<font color="Maroon" title="c2">PS</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font>
 <b>and </b><br/><a NAME="E8:13_1_2_1_1"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c5">W</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <font color="Maroon" title="c6">V</font>
 <b>by </b><i><a class="txt" href="compts_1.html#E1:13"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="compts_1.html#E2:13"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="compts_1.html#E1:13_1_2"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="compts_1.html#E1:13_1_2_1_1"><i><font color="Green" title="E16">A4</font></i></a>, <a class="ref" href="compts_1.html#T15" target="_self" title="COMPTS_1:th.15">Th15</a></i>;<br/>
<a NAME="E9:13_1_2_1_1"/>
<font color="Maroon" title="c5">W</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> </span>)</span> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a> 
 
<b>by </b><i><a class="txt" href="compts_1.html#E8:13_1_2_1_1"><i><font color="Green" title="E20">A8</font></i></a></i>;<br/>
<a NAME="E10:13_1_2_1_1"/><b>then </b><i><font color="Green" title="E21">A9</font></i>: 
( <font color="Maroon" title="c5">W</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c6">V</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">PS</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 
<b>by </b><i><a class="txt" href="compts_1.html#E7:13_1_2_1_1"><i><font color="Green" title="E19">A7</font></i></a>, <a class="ref" href="subset_1.html#T31" title="SUBSET_1:th.31">SUBSET_1:31</a>, <a class="ref" href="subset_1.html#T44" title="SUBSET_1:th.44">SUBSET_1:44</a></i>;<br/>
<b>take </b><font color="Maroon" title="c7">Q</font> = <font color="Maroon" title="c5">W</font>;<br/>

<b>thus </b><a NAME="E11:13_1_2_1_1"/>
( <font color="Maroon" title="c7">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c7">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">PS</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q</font> )
 <b>by </b><i><a class="txt" href="compts_1.html#E4:13_1_2_1_1"><i><font color="Green" title="E17">A5</font></i></a>, <a class="txt" href="compts_1.html#E6:13_1_2_1_1"><i><font color="Green" title="E18">A6</font></i></a>, <a class="txt" href="compts_1.html#E10:13_1_2_1_1"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E2:13_1_2_1"/>
(  ex <font color="Olive" title="b1">Q</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">TS</font> st <br/>( <font color="Olive" title="b1">Q</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Olive" title="b1">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">PS</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  &amp; <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">Q</font> ) implies <font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">PS</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  )
 ;<br/></div><b>end;</b></div><a NAME="E3:13_1_2"/><b>then </b>
<font color="Maroon" title="c2">PS</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T57" title="TOPS_1:th.57">TOPS_1:57</a></i>;<br/><b>hence </b><a NAME="E4:13_1_2"/>
<font color="Maroon" title="c2">PS</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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