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<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="E1:51"/>
( <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="tops_1.html#V5">closed_condensed</a> &amp; <font color="Olive">A</font> is <a href="tops_1.html#V6">open_condensed</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_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="E2:51_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="E3:51_1"/><b>then </b>
( <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>, <a class="ref" href="tdlat_3.html#T11" target="_self">Th11</a></i>;<br/>
<b>hence </b><a NAME="E4:51_1"/>
( <font color="Maroon">A</font> is <a href="tops_1.html#V5">closed_condensed</a> &amp; <font color="Maroon">A</font> is <a href="tops_1.html#V6">open_condensed</a> )
 <b>by </b><i><a class="ref" href="tdlat_3.html#D2" target="_self">Def2</a>, <a class="ref" href="tops_1.html#T106">TOPS_1:106</a>, <a class="ref" href="tops_1.html#T107">TOPS_1:107</a></i>;<br/>


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

<b>assume </b><a NAME="E2:51"/><i><font color="Green">E45</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="tops_1.html#V5">closed_condensed</a> &amp; <font color="Olive">A</font> is <a href="tops_1.html#V6">open_condensed</a> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_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="E1:51_2"/>
<font color="Maroon">A</font> is <a href="tops_1.html#V4">condensed</a>
 ;<br/><a NAME="E2:51_2"/><b>then </b>
( <font color="Maroon">A</font> is <a href="tops_1.html#V5">closed_condensed</a> &amp; <font color="Maroon">A</font> is <a href="tops_1.html#V6">open_condensed</a> )
 <b>by </b><i><a class="ref" href="tdlat_3.html#D1" target="_self">Def1</a></i>;<br/><b>hence </b><a NAME="E3:51_2"/>
( <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="tops_1.html#T106">TOPS_1:106</a>, <a class="ref" href="tops_1.html#T107">TOPS_1:107</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:51"/>
<font color="Maroon">X</font> is <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a>
 <b>by </b><i><a class="ref" href="tdlat_3.html#T11" target="_self">Th11</a></i>;<br/>


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