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

<b>let </b><font color="Maroon" title="c1">T</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">F</font> is <a href="tdlat_2.html#V3" title="TDLAT_2:attr.3">open-domains-family</a> holds <br/><font color="Olive" title="b1">F</font> is <a href="tdlat_2.html#V1" title="TDLAT_2:attr.1">domains-family</a></span><br/><b>let </b><font color="Maroon" title="c2">F</font> be   <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is <a href="tdlat_2.html#V3" title="TDLAT_2:attr.3">open-domains-family</a> implies <font color="Maroon" title="c2">F</font> is <a href="tdlat_2.html#V1" title="TDLAT_2:attr.1">domains-family</a> )</span><br/>



<b>thus </b><a NAME="E1:85"/>
( <font color="Maroon" title="c2">F</font> is <a href="tdlat_2.html#V3" title="TDLAT_2:attr.3">open-domains-family</a> implies <font color="Maroon" title="c2">F</font> is <a href="tdlat_2.html#V1" title="TDLAT_2:attr.1">domains-family</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><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" title="c2">F</font> is <a href="tdlat_2.html#V3" title="TDLAT_2:attr.3">open-domains-family</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">F</font> is <a href="tdlat_2.html#V1" title="TDLAT_2:attr.1">domains-family</a></span><br/>

<a NAME="E2:85_1"/><b>then </b>
( <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tdlat_1.html#K9" title="TDLAT_1:func.9">Open_Domains_of</a> <font color="Maroon" title="c1">T</font> &amp;  <a href="tdlat_1.html#K9" title="TDLAT_1:func.9">Open_Domains_of</a> <font color="Maroon" title="c1">T</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tdlat_1.html#K1" title="TDLAT_1:func.1">Domains_of</a> <font color="Maroon" title="c1">T</font> )
 
<b>by </b><i><a class="ref" href="tdlat_2.html#T79" target="_self" title="TDLAT_2:th.79">Th79</a>, <a class="ref" href="tdlat_1.html#T35" title="TDLAT_1:th.35">TDLAT_1:35</a></i>;<br/>
<a NAME="E3:85_1"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="tdlat_1.html#K1" title="TDLAT_1:func.1">Domains_of</a> <font color="Maroon" title="c1">T</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E4:85_1"/>
<font color="Maroon" title="c2">F</font> is <a href="tdlat_2.html#V1" title="TDLAT_2:attr.1">domains-family</a>
 <b>by </b><i><a class="ref" href="tdlat_2.html#T65" target="_self" title="TDLAT_2:th.65">Th65</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


</div>
