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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/>

<b>assume </b><a NAME="E1:75"/>
 <a href="tdlat_1.html#K8">Closed_Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="tdlat_1.html#K12">Open_Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E2:75"/><b>then </b>
 <a href="tdlat_1.html#K5">Closed_Domains_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="tdlat_1.html#K12">Open_Domains_Lattice</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T94">TDLAT_2:94</a></i>;<br/>
<a NAME="E3:75"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="tdlat_1.html#K5">Closed_Domains_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="tdlat_1.html#K9">Open_Domains_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T103">TDLAT_2:103</a></i>;<br/>
<a NAME="E4:75"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( ( <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V6">open_condensed</a> implies <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V5">closed_condensed</a> ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V5">closed_condensed</a> implies <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V6">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">c<sub>2</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>thus </b><a NAME="E1:75_1"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V6">open_condensed</a> implies <font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V5">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">c<sub>2</sub></font> is <a href="tops_1.html#V6">open_condensed</a>
 ;<br/>

<a NAME="E2:75_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V6">open_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E3:75_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K5">Closed_Domains_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="tdlat_3.html#E3:75"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tdlat_1.html#D9">TDLAT_1:def 9</a></i>;<br/>
<a NAME="E4:75_1_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D5">TDLAT_1:def 5</a></i>;<br/>
<a NAME="E5:75_1_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V5">closed_condensed</a> )
 
;<br/>
<b>hence </b><a NAME="E6:75_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V5">closed_condensed</a>
 ;<br/>


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

<b>assume </b><a NAME="E2:75_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V5">closed_condensed</a>
 ;<br/>

<a NAME="E3:75_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V5">closed_condensed</a> </span>}</span> 
 
;<br/>
<a NAME="E4:75_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="tdlat_1.html#K9">Open_Domains_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="tdlat_3.html#E3:75"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tdlat_1.html#D5">TDLAT_1:def 5</a></i>;<br/>
<a NAME="E5:75_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V6">open_condensed</a> </span>}</span> 
 
<b>by </b><i><a class="ref" href="tdlat_1.html#D9">TDLAT_1:def 9</a></i>;<br/>
<a NAME="E6:75_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> is <a href="tops_1.html#V6">open_condensed</a> )
 
;<br/>
<b>hence </b><a NAME="E7:75_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_1.html#V6">open_condensed</a>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:75"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="tdlat_3.html#V4" target="_self">extremally_disconnected</a>
 <b>by </b><i><a class="ref" href="tdlat_3.html#T38" target="_self">Th38</a></i>;<br/>


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