<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:75"/>
 <a href="tdlat_1.html#K8" title="TDLAT_1:func.8">Closed_Domains_Lattice</a> <font color="Maroon" title="c1">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="tdlat_1.html#K12" title="TDLAT_1:func.12">Open_Domains_Lattice</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E2:75"/>
for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> holds <br/> ( ( <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> implies <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> ) &amp; ( <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> implies <font color="Olive" title="b1">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font>;<br/>

<b>thus </b><a NAME="E1:75_1"/>
( <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> implies <font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:75_1_1"/>
<font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a>
 ;<br/>

<a NAME="E2:75_1_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">E</font> where <font color="Olive" title="b1">E</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">E</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E3:75_1_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">E</font> where <font color="Olive" title="b1">E</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">E</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E4:75_1_1"/><b>then </b>
 ex <font color="Olive" title="b1">D</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">D</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> )
 
;<br/>
<b>hence </b><a NAME="E5:75_1_1"/>
<font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a>
 ;<br/>


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

<b>assume </b><a NAME="E2:75_1"/>
<font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a>
 ;<br/>

<a NAME="E3:75_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">E</font> where <font color="Olive" title="b1">E</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">E</font> is <a href="tops_1.html#V5" title="TOPS_1:attr.5">closed_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E4:75_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">E</font> where <font color="Olive" title="b1">E</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> : <font color="Olive" title="b1">E</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E5:75_1"/><b>then </b>
 ex <font color="Olive" title="b1">D</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">X</font> st <br/>( <font color="Olive" title="b1">D</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Olive" title="b1">D</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a> )
 
;<br/>
<b>hence </b><a NAME="E6:75_1"/>
<font color="Maroon" title="c2">A</font> is <a href="tops_1.html#V6" title="TOPS_1:attr.6">open_condensed</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:75"/>
<font color="Maroon" title="c1">X</font> is <a href="tdlat_3.html#V4" title="TDLAT_3:attr.4">extremally_disconnected</a>
 <b>by </b><i><a class="ref" href="tdlat_3.html#T38" target="_self" title="TDLAT_3:th.38">Th38</a></i>;<br/>


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