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<div class="add">

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>thus </b><a NAME="E1:50"/>
( <font color="Maroon">X</font> is <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a> implies for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A</font> is <a href="tops_1.html#V4">condensed</a> holds <br/>( <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:50_1"/><i><font color="Green">E21</font></i>: 
<font color="Maroon">X</font> is <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a>
 ;<br/>

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/>

<b>assume </b><a NAME="E2:50_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">A</font> is <a href="tops_1.html#V4">condensed</a>
 ;<br/>

<a NAME="E3:50_1"/><b>then </b><i><font color="Green">E45</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E4:50_1"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="txt" href="tdlat_3.html#E1:50_1"><i><font color="Green">E21</font></i></a></i>;<br/>
<a NAME="E5:50_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_3.html#D2" target="_self">Def2</a>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/>
<b>hence </b><a NAME="E6:50_1"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="tdlat_3.html#D1" target="_self">Def1</a>, </i>;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E2:50"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>  st <font color="Olive">A</font> is <a href="tops_1.html#V4">condensed</a> holds <br/>( <font color="Olive">A</font> is <a href="pre_topc.html#V4">closed</a> &amp; <font color="Olive">A</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="50_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>assume </b><a NAME="E1:50_2"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/><a NAME="E2:50_2"/><b>then </b>
(  <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span> is <a href="tops_1.html#V5">closed_condensed</a> )
 <b>by </b><i><a class="ref" href="tdlat_1.html#T22">TDLAT_1:22</a>, <a class="ref" href="tops_1.html#T55">TOPS_1:55</a></i>;<br/><a NAME="E3:50_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> is <a href="tops_1.html#V4">condensed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T106">TOPS_1:106</a></i>;<br/><b>hence </b><a NAME="E4:50_2"/>
 <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:50"/>
<font color="Maroon">X</font> is <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a>
 <b>by </b><i/>;<br/>


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