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

<b>let </b><font color="Maroon">X</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">A</font> be   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font>;<br/>



<b>set </b><font color="Maroon">T</font> =  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #);<br/>
<b>reconsider </b><font color="Maroon">A'</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM3">Subset-Family</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #) ;<br/>
<a NAME="E2:26"/>
<font color="Maroon">A'</font> is    <a href="cantor_1.html#M2" target="_self">prebasis</a> of  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #)
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>proof </b></a><div class="add">

<a NAME="E1:26_1"/>
( <font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">A</font> &amp;  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">A</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E2:26_1"/>
<font color="Maroon">A'</font> <a href="tarski.html#R1">c=</a> the <a href="pre_topc.html#U1">topology</a> of <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #)
 <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="cantor_1.html#D5" target="_self">CANTOR_1:def 5</a><br/>

<b>reconsider </b><font color="Maroon">B</font> =  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">A'</font> as    <a href="cantor_1.html#M1" target="_self">Basis</a> of  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #) <b>by </b><i><a class="ref" href="cantor_1.html#D1" target="_self">Def1</a></i>;<br/>
<b>take </b>
<font color="Maroon">B</font>
;<br/>

<b>thus </b><a NAME="E4:26_1"/>
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">A'</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:26"/>
<font color="Maroon">A</font> is    <a href="cantor_1.html#M2" target="_self">prebasis</a> of  <a href="pre_topc.html#G1">TopStruct</a>(# <font color="Maroon">X</font>,<span class="p1">(<span class="default"><a href="cantor_1.html#K1" target="_self">UniCl</a> <span class="p2">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">A</font></span>)</span></span>)</span> #)
 ;<br/>


</div>
