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

<b>let </b><font color="Maroon">X</font> be    <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>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="E1:19"/><i><font color="Green">E23</font></i>: 
( <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; <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/>

<b>let </b><font color="Maroon">Z</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="E2:19"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K3">INTERSECTION</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon">V</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">B</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:19"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">V</font> <a href="hidden.html#R2">in</a> <font color="Maroon">a</font>
 <b>and </b><br/><a NAME="E5:19"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">B</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font>
 <b>and </b><br/><a NAME="E6:19"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">Z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="setfam_1.html#D5">SETFAM_1:def 5</a></i>;<br/>
<a NAME="E7:19"/>
( <font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">A</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>then reconsider </b><font color="Maroon">M</font> = <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">V</font>,<font color="Maroon">B</font></span><a href="tarski.html#K2">}</a></span> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">M</font> = <font color="Maroon">M</font> as   <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> ;<br/>
<a NAME="E10:19"/>
<font color="Maroon">V</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K6">meet</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#T12">SETFAM_1:12</a></i>;<br/>
<a NAME="E11:19"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">V</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">B</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>;<br/>
<a NAME="E12:19"/>
( <font color="Maroon">M</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="setfam_1.html#NM1">Subset-Family</a> of <font color="Maroon">X</font> &amp; <font color="Maroon">M</font> <a href="tarski.html#R1">c=</a>  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">A</font> )
 
<b>by </b><i>, , , <a class="ref" href="zfmisc_1.html#T38">ZFMISC_1:38</a></i>;<br/>
<a NAME="E13:19"/><b>then </b>
 <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">M</font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <span class="p1">(<span class="default"><a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">A</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E14:19"/>
<font color="Maroon">Z</font> <a href="hidden.html#R2">in</a>  <a href="cantor_1.html#K2" target="_self">FinMeetCl</a> <font color="Maroon">A</font>
 <b>by </b><i><a class="ref" href="cantor_1.html#D1" target="_self">Def1</a>, , </i>;<br/>


</div>
