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

<b>set </b><font color="Maroon">X</font> = <font color="Maroon">F<sub>1</sub></font>();<br/>
<b>set </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] </span>}</span> ;<br/>
<a NAME="E1:6_1"/>
<span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:6_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] </span>}</span> 
 ;<br/>

<a NAME="E2:6_1_1"/><b>then </b>
 ex <font color="Olive">B</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">B</font>] )
 
;<br/>
<b>hence </b><a NAME="E3:6_1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="zfmisc_1.html#K1">bool</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">F</font> = <span class="p1">{<span class="default"> <font color="Olive">A</font> where <font color="Olive">A</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() : <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">A</font>] </span>}</span>  as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>() ;<br/>
<b>set </b><font color="Maroon">T</font> =  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font></span>)</span> #);<br/>
<a NAME="E3:6_1"/>
(  <a href="arytm_3.html#K13">{}</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>() &amp; <font color="Maroon">F<sub>1</sub></font>() <a href="tarski.html#R1">c=</a> <font color="Maroon">F<sub>1</sub></font>() )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T2">XBOOLE_1:2</a></i>;<br/>
<a NAME="E4:6_1"/><b>then </b><i><font color="Green">E44</font></i>: 
(  <a href="arytm_3.html#K13">{}</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F<sub>1</sub></font>() <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E5:6_1"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">A</font>, <font color="Olive">B</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> holds <br/><font color="Olive">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Olive">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_2"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:6_1_2"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">A'</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E3:6_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A'</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A'</font>] )
 ;<br/>
<b>assume </b><a NAME="E4:6_1_2"/>
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>

<b>then consider </b><font color="Maroon">B'</font> being   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>()<b> such that </b><br/><a NAME="E6:6_1_2"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">B'</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">B'</font>] )
 ;<br/>
<a NAME="E7:6_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A'</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B'</font>]
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E8:6_1_2"/>
<font color="Maroon">A</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:6_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">G</font> being  <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>()  st <font color="Olive">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> holds <br/> <a href="setfam_1.html#K8">Intersect</a> <font color="Olive">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">G</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">F<sub>1</sub></font>();<br/>

<b>assume </b><a NAME="E1:6_1_3"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_3_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">A</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:6_1_3_1"/>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">G</font>
 ;<br/><a NAME="E2:6_1_3_1"/><b>then </b>
<font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/><a NAME="E3:6_1_3_1"/><b>then </b>
 ex <font color="Olive">B</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">B</font>] )
 ;<br/><b>hence </b><a NAME="E4:6_1_3_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:6_1_3"/><b>then </b>
<font color="Maroon">P<sub>1</sub></font>[ <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">G</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:6_1_3"/>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">G</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">T</font> =  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">F<sub>1</sub></font>(),<span class="p1">(<span class="default"><a href="setfam_1.html#K7">COMPLEMENT</a> <font color="Maroon">F</font></span>)</span> #) as  <a href="pre_topc.html#V1">strict</a> <a href="pre_topc.html#NM1">TopSpace</a> <b>by </b><i>, , </i>;<br/>
<b>take </b>
<font color="Maroon">T</font>
;<br/>

<b>thus </b><a NAME="E8:6_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 ;<br/>

<b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>

<a NAME="E9:6_1"/>
( ( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> implies  ex <font color="Olive">B</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">B</font>] ) ) &amp; (  ex <font color="Olive">B</font> being  <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">F<sub>1</sub></font>() st <br/>( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Olive">B</font> &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive">B</font>] ) implies <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> ) &amp; ( <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> implies <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> ) &amp; ( <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> implies <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> ) )
 
<b>by </b><i>, , , </i>;<br/>
<b>hence </b><a NAME="E10:6_1"/>
( <font color="Maroon">A</font> is <a href="pre_topc.html#V4">closed</a> iff <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon">A</font>] )
 ;<br/>


</div>
