<?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">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">FX</font> be   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font>;<br/>





<b>assume </b><a NAME="E1:14"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">FX</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">V</font></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>reconsider </b><font color="Maroon">CFX</font> =  <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> as    <a href="hidden.html#M1">set</a>  ;<br/>
<a NAME="E3:14"/>
for <font color="Olive">W</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <font color="Olive">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CFX</font> iff <font color="Olive">W</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:14_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CFX</font> implies <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CFX</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">W</font> = <font color="Maroon">W</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E3:14_1_1"/>
 ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:14_1_1"/>
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:14_1"/>
( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">V</font> implies <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CFX</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:14_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">V</font>
 ;<br/>

<a NAME="E2:14_1_2"/>
 ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Olive">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_2_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon">V</font>
;<br/>

<b>thus </b><a NAME="E1:14_1_2_1"/>
( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FX</font> )
 <b>by </b><i>, , <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:14_1_2"/>
<font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CFX</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:14_1"/>
( <font color="Maroon">W</font> <a href="hidden.html#R2">in</a> <font color="Maroon">CFX</font> iff <font color="Maroon">W</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <font color="Maroon">V</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:14"/>
 <a href="pcomps_1.html#K2" target="_self">clf</a> <font color="Maroon">FX</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <font color="Maroon">V</font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


</div>
