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

<b>let </b><font color="Maroon" title="c1">T</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font> be   <a href="setfam_1.html#NM1" title="SETFAM_1:NM.1">Subset-Family</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>set </b><font color="Maroon" title="c4">XX</font> = <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = <span class="p1">(<span class="default">$1 <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> </span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default">$1 <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a> </span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c5">IN</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">X</font>) where <font color="Olive" title="b1">X</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> </span>}</span> ;<br/>
<b>set </b><font color="Maroon" title="c6">A</font> = <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>;<br/>
<b>set </b><font color="Maroon" title="c7">AL</font> = <font color="Maroon" title="c2">F</font>;<br/>
<b>set </b><font color="Maroon" title="c8">BL</font> = <font color="Maroon" title="c3">G</font>;<br/>
<a NAME="E1:46"/><i><font color="Green" title="E25">A1</font></i>: 
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">w</font>) where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> : <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> </span>}</span>  <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>from </b><i><a class="ref" href="trees_2.html#S2" title="TREES_2:sch.2">TREES_2:sch 2</a>();<br/></i>
<b>set </b><font color="Maroon" title="c9">C</font> =  <a href="topgen_4.html#K2" title="TOPGEN_4:func.2">INTERSECTION</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font>;<br/>
<a NAME="E2:46"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">X</font>) where <font color="Olive" title="b1">X</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topgen_4.html#K2" title="TOPGEN_4:func.2">INTERSECTION</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:46_1"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">X</font>) where <font color="Olive" title="b1">X</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topgen_4.html#K2" title="TOPGEN_4:func.2">INTERSECTION</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:46_1_1"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">X</font>) where <font color="Olive" title="b1">X</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">X</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span><b> such that </b><br/><a NAME="E3:46_1_1"/><i><font color="Green" title="E26">A2</font></i>: 
( <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">X</font>) &amp; <font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> )
 ;<br/>
<a NAME="E4:46_1_1"/><i><font color="Green" title="E27">A3</font></i>: 
( <font color="Maroon" title="c11">X</font> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c11">X</font> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> )
 
<b>by </b><i><a class="txt" href="topgen_4.html#E3:46_1_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="mcart_1.html#T10" title="MCART_1:th.10">MCART_1:10</a></i>;<br/>
<b>thus </b><a NAME="E5:46_1_1"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_4.html#K2" title="TOPGEN_4:func.2">INTERSECTION</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="txt" href="topgen_4.html#E3:46_1_1"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="topgen_4.html#E4:46_1_1"><i><font color="Green" title="E27">A3</font></i></a>, <a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>


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

<b>thus </b><a NAME="E2:46_1"/>
 <a href="topgen_4.html#K2" title="TOPGEN_4:func.2">INTERSECTION</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">X</font>) where <font color="Olive" title="b1">X</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:46_1_2"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topgen_4.html#K2" title="TOPGEN_4:func.2">INTERSECTION</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">X</font>, <font color="Maroon" title="c12">Y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:46_1_2"/><i><font color="Green" title="E26">A4</font></i>: 
( <font color="Maroon" title="c11">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c12">Y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">G</font> &amp; <font color="Maroon" title="c10">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">X</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <font color="Maroon" title="c12">Y</font> )
 <b>by </b><i><a class="ref" href="setfam_1.html#D5" title="SETFAM_1:def.5">SETFAM_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">X</font> = <font color="Maroon" title="c11">X</font>, <font color="Maroon" title="c14">Y</font> = <font color="Maroon" title="c12">Y</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="txt" href="topgen_4.html#E3:46_1_2"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">XY</font> = <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">X</font>,<font color="Maroon" title="c14">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span>;<br/>
<a NAME="E5:46_1_2"/><i><font color="Green" title="E27">A5</font></i>: 
( <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">X</font>,<font color="Maroon" title="c14">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="domain_1.html#K2" title="DOMAIN_1:func.2">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">X</font> &amp; <span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">X</font>,<font color="Maroon" title="c14">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="domain_1.html#K3" title="DOMAIN_1:func.3">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">Y</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7" title="MCART_1:th.7">MCART_1:7</a></i>;<br/>
<a NAME="E6:46_1_2"/>
<span class="p1"><a href="domain_1.html#K1" title="DOMAIN_1:func.1">[</a><span class="default"><font color="Maroon" title="c13">X</font>,<font color="Maroon" title="c14">Y</font></span><a href="domain_1.html#K1" title="DOMAIN_1:func.1">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 
<b>by </b><i><a class="txt" href="topgen_4.html#E3:46_1_2"><i><font color="Green" title="E26">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>hence </b><a NAME="E7:46_1_2"/>
<font color="Maroon" title="c10">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">X</font>) where <font color="Olive" title="b1">X</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span>,<span class="p2">(<span class="default"><a href="pcomps_1.html#K1" title="PCOMPS_1:func.1">bool</a> <font color="Maroon" title="c1">T</font></span>)</span></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> : <font color="Olive" title="b1">X</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span> </span>}</span> 
 <b>by </b><i><a class="txt" href="topgen_4.html#E3:46_1_2"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="topgen_4.html#E5:46_1_2"><i><font color="Green" title="E27">A5</font></i></a></i>;<br/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:46"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1">(<span class="default"><a href="topgen_4.html#K2" title="TOPGEN_4:func.2">INTERSECTION</a> <font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span>)</span> <a href="card_1.html#NR1" title="CARD_1:NR.1">&lt;=`</a>  <a href="card_1.html#K1" title="CARD_1:func.1">Card</a> <span class="p1"><a href="mcart_1.html#K12" title="MCART_1:func.12">[:</a><span class="default"><font color="Maroon" title="c2">F</font>,<font color="Maroon" title="c3">G</font></span><a href="mcart_1.html#K12" title="MCART_1:func.12">:]</a></span>
 <b>by </b><i><a class="txt" href="topgen_4.html#E1:46"><i><font color="Green" title="E25">A1</font></i></a></i>;<br/>


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