<?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="E1:84"/>
( <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> implies <font color="Maroon">c<sub>2</sub></font> is <a href="tdlat_2.html#V3" target="_self">open-domains-family</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_1"/><i><font color="Green">E49</font></i>: 
<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>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_1_1"><b>now </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="E1:84_1_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="E2:84_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</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_2.html#E1:84_1"><i><font color="Green">E49</font></i></a></i>;<br/><a NAME="E3:84_1_1"/><b>then </b>
<font color="Maroon">c<sub>3</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="E4:84_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>3</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="E5:84_1_1"/>
<font color="Maroon">c<sub>3</sub></font> is <a href="tops_1.html#V6">open_condensed</a>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:84_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="tdlat_2.html#V3" target="_self">open-domains-family</a>
 <b>by </b><i><a class="ref" href="tdlat_2.html#D4" target="_self">Def4</a></i>;<br/>


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

<b>thus </b><a NAME="E2:84"/>
( <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> <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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:84_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="tdlat_2.html#V3" target="_self">open-domains-family</a>
 ;<br/>

<a NAME="E2:84_2"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  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> <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>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="84_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:84_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E3:84_2_1"/>
<font color="Maroon">c<sub>4</sub></font> is <a href="tops_1.html#V6">open_condensed</a>
 
<b>by </b><i><a class="txt" href="tdlat_2.html#E1:84_2"><i><font color="Green">E49</font></i></a>, <a class="txt" href="tdlat_2.html#E1:84_2_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="tdlat_2.html#D4" target="_self">Def4</a></i>;<br/>
<a NAME="E4:84_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</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/>
<b>hence </b><a NAME="E5:84_2_1"/>
<font color="Maroon">c<sub>3</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="ref" href="tdlat_1.html#D9">TDLAT_1:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:84_2"/>
<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>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


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


</div>
