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



<b>assume </b><a NAME="E1:59"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">A</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>  st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Olive">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Olive">A</font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E2:59"/><i><font color="Green">E19</font></i>: 
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A0</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:59_1_1"/><i><font color="Green">E22</font></i>: 
<font color="Maroon">A0</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/><b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A0</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/><a NAME="E3:59_1_1"/>
<font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font>
 <b>by </b><i>, <a class="ref" href="zfmisc_1.html#T92">ZFMISC_1:92</a></i>;<br/><a NAME="E4:59_1_1"/><b>then </b>
 <a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span>
 <b>by </b><i><a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/><a NAME="E5:59_1_1"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span></span>)</span> &amp; <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <font color="Maroon">A</font></span>)</span> )
 <b>by </b><i>, , <a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/><b>hence </b><a NAME="E6:59_1_1"/>
<font color="Maroon">A0</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:59_1"/>
 <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T94">ZFMISC_1:94</a></i>;<br/>


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

<b>thus </b><a NAME="E3:59"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>proof </b></a><div class="add">

<a NAME="E1:59_2"/>
(  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span></span>)</span> &amp;  <a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="pre_topc.html#T48">PRE_TOPC:48</a></i>;<br/>
<a NAME="E2:59_2"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> &amp;  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a>, <a class="ref" href="tops_1.html#T48">TOPS_1:48</a></i>;<br/>
<a NAME="E3:59_2"/><b>then </b>
(  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span></span>)</span> &amp;  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a>, <a class="ref" href="tops_1.html#T26">TOPS_1:26</a></i>;<br/>
<a NAME="E4:59_2"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<a NAME="E5:59_2"/>
 <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T44">TOPS_1:44</a></i>;<br/>
<a NAME="E6:59_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p2">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T49">PRE_TOPC:49</a></i>;<br/>
<a NAME="E7:59_2"/><b>then </b>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span>
 
<b>by </b><i><a class="ref" href="tops_1.html#T26">TOPS_1:26</a></i>;<br/>
<b>hence </b><a NAME="E8:59_2"/>
 <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K4">Cl</a> <span class="p1">(<span class="default"><a href="tops_1.html#K1">Int</a> <span class="p2">(<span class="default"><a href="pre_topc.html#K4">Cl</a> <span class="p3">(<span class="default"><a href="setfam_1.html#K5">union</a> <font color="Maroon">F</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


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


</div>
