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<b>let </b><font color="Maroon">TS</font> be   <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">PS</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:13"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">TS</font> <a href="compts_1.html#NR2" target="_self">is_T2</a> 
 <b>and </b><br/><a NAME="E2:13"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">PS</font> is <a href="compts_1.html#V6" target="_self">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">PS</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">PS</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</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">PS</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><a NAME="E2:13_1_1"/><b>then </b>
<font color="Maroon">PS</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <font color="Maroon">TS</font>
 ;<br/><b>hence </b><a NAME="E3:13_1_1"/>
<font color="Maroon">PS</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T22">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">E34</font></i>: 
<font color="Maroon">PS</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</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">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>thus </b><a NAME="E1:13_1_2_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PS</font> <a href="subset_1.html#K3">`</a>  implies  ex <font color="Olive">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font> st <br/>( <font color="Olive">Q</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PS</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">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">E35</font></i>: 
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PS</font> <a href="subset_1.html#K3">`</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">a</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">TS</font> ;<br/>
<b>consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font>, <font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font><b> such that </b><br/><a NAME="E4:13_1_2_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">W</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E5:13_1_2_1_1"/>
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 <b>and </b><br/><a NAME="E6:13_1_2_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font>
 <b>and </b><br/><a NAME="E7:13_1_2_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">PS</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 <b>and </b><br/><a NAME="E8:13_1_2_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a> <font color="Maroon">V</font>
 <b>by </b><i>, , , , </i>;<br/>
<a NAME="E9:13_1_2_1_1"/>
<font color="Maroon">W</font> <a href="xboole_0.html#R1">misses</a> <span class="p1">(<span class="default"><font color="Maroon">V</font> <a href="subset_1.html#K3">`</a> </span>)</span> <a href="subset_1.html#K3">`</a> 
 
<b>by </b><i><a class="ref" href="compts_1.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E10:13_1_2_1_1"/><b>then </b><i><font color="Green">E40</font></i>: 
( <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">V</font> <a href="subset_1.html#K3">`</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">PS</font> <a href="subset_1.html#K3">`</a>  )
 
<b>by </b><i><a class="ref" href="compts_1.html#D1" target="_self">Def1</a>, <a class="ref" href="subset_1.html#T31">SUBSET_1:31</a>, <a class="ref" href="subset_1.html#T44">SUBSET_1:44</a></i>;<br/>
<b>take </b><font color="Maroon">Q</font> = <font color="Maroon">W</font>;<br/>

<b>thus </b><a NAME="E11:13_1_2_1_1"/>
( <font color="Maroon">Q</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PS</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font> )
 <b>by </b><i>, , , <a class="ref" href="xboole_1.html#T1">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">Q</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">TS</font> st <br/>( <font color="Olive">Q</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Q</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">PS</font> <a href="subset_1.html#K3">`</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">Q</font> ) implies <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">PS</font> <a href="subset_1.html#K3">`</a>  )
 ;<br/></div><b>end;</b></div><a NAME="E3:13_1_2"/><b>then </b>
<font color="Maroon">PS</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T57">TOPS_1:57</a></i>;<br/><b>hence </b><a NAME="E4:13_1_2"/>
<font color="Maroon">PS</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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