<?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>thus </b><a NAME="E1:85"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="tdlat_2.html#V3" target="_self">open-domains-family</a> implies <font color="Maroon">c<sub>2</sub></font> is <a href="tdlat_2.html#V1" target="_self">domains-family</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="85_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:85_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tdlat_2.html#V3" target="_self">open-domains-family</a>
 ;<br/>

<a NAME="E2:85_1"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tdlat_1.html#K9">Open_Domains_of</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="tdlat_1.html#K9">Open_Domains_of</a> <font color="Maroon">c<sub>1</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tdlat_1.html#K1">Domains_of</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T79" target="_self">Th79</a>, <a class="ref" href="tdlat_1.html#T35">TDLAT_1:35</a></i>;<br/>
<a NAME="E3:85_1"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="tdlat_1.html#K1">Domains_of</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E4:85_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tdlat_2.html#V1" target="_self">domains-family</a>
 <b>by </b><i><a class="ref" href="tdlat_2.html#T65" target="_self">Th65</a></i>;<br/>


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


</div>
