<?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>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:78"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tdlat_2.html#V2" target="_self">closed-domains-family</a>
 ;<br/>

<a NAME="E2:78"/>
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> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="pre_topc.html#V4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:78_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<a NAME="E2:78_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V5">closed_condensed</a>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E1:78"><i><font color="Green">E44</font></i></a>, <a class="ref" href="tdlat_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E3:78_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T106">TOPS_1:106</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:78"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tops_2.html#V2">closed</a>
 <b>by </b><i><a class="ref" href="tops_2.html#D2">TOPS_2:def 2</a></i>;<br/>


</div>
