<?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">B</font> be    <a href="cantor_1.html#M1">Basis</a> of <font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>





<b>set </b><font color="Maroon">Bx</font> = <span class="p1">{<span class="default"> <font color="Olive">U</font> where <font color="Olive">U</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> &amp; <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span> ;<br/>
<a NAME="E1:1"/><i><font color="Green">E29</font></i>: 
<span class="p1">{<span class="default"> <font color="Olive">U</font> where <font color="Olive">U</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> &amp; <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="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="E1:1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">U</font> where <font color="Olive">U</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> &amp; <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span> 
 ;<br/>

<a NAME="E2:1_1"/><b>then </b>
 ex <font color="Olive">U</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">U</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> &amp; <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 
;<br/>
<b>hence </b><a NAME="E3:1_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:1"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">U</font> where <font color="Olive">U</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> &amp; <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="pcomps_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">Bx</font> = <span class="p1">{<span class="default"> <font color="Olive">U</font> where <font color="Olive">U</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> &amp; <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span>  as   <a href="struct_0.html#NM3">Subset-Family</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E4:1"/>
<font color="Maroon">Bx</font> is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2"><b>proof </b></a><div class="add">

<a NAME="E1:1_2"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="cantor_1.html#D2">CANTOR_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E2:1_2"/>
<font color="Maroon">Bx</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="yellow_8.html#D2">YELLOW_8:def 2</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_2_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="E1:1_2_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bx</font>
 ;<br/><a NAME="E2:1_2_1"/><b>then </b>
 ex <font color="Olive">U</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> st <br/>( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">U</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> &amp; <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> )
 ;<br/><b>hence </b><a NAME="E3:1_2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Bx</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#T58">SETFAM_1:58</a></i>;<br/>

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

<b>assume </b><a NAME="E5:1_2"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 ;<br/>

<b>consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font><b> such that </b><br/><a NAME="E7:1_2"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 <b>by </b><i>, <a class="ref" href="yellow_9.html#T31">YELLOW_9:31</a></i>;<br/>
<a NAME="E8:1_2"/>
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bx</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:1_2"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of <a href="zfmisc_1.html#K1">K10</a>(the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>) st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">Bx</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">S</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:1"/>
<span class="p1">{<span class="default"> <font color="Olive">U</font> where <font color="Olive">U</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">U</font> &amp; <font color="Olive">U</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font> ) </span>}</span>  is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">x</font>
 ;<br/>


</div>
