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

<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:79"/><i><font color="Green">E18</font></i>: 
<font color="Maroon">F</font> is <a href="tdlat_2.html#V1" target="_self">domains-family</a>
 ;<br/>

<a NAME="E2:79"/>
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font> holds <br/><font color="Olive">A</font> is <a href="tops_1.html#V5">closed_condensed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </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">T</font>;<br/>

<b>assume </b><a NAME="E1:79_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E3:79_1"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">P</font>
 <b>and </b><br/><a NAME="E4:79_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i><a class="ref" href="pcomps_1.html#D3">PCOMPS_1:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P</font> = <font color="Maroon">P</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E6:79_1"/>
<font color="Maroon">P</font> is <a href="tops_1.html#V4">condensed</a>
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T13" target="_self">Th13</a>, <a class="ref" href="tdlat_2.html#T14" target="_self">Th14</a>, </i>;<br/>
<b>hence </b><a NAME="E7:79_1"/>
<font color="Maroon">A</font> is <a href="tops_1.html#V5">closed_condensed</a>
 <b>by </b><i>, <a class="ref" href="tdlat_1.html#T24">TDLAT_1:24</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:79"/>
 <a href="tdlat_2.html#NK1" target="_self">Cl</a> <font color="Maroon">F</font> is <a href="tdlat_2.html#V2" target="_self">closed-domains-family</a>
 <b>by </b><i><a class="ref" href="tdlat_2.html#T9" target="_self">Th9</a></i>;<br/>


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