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

<b>let </b><font color="Maroon">I</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">J</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">F</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">I</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/><b>let </b><font color="Maroon">G</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">J</font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>





<a NAME="E1:20"/><i><font color="Green">E25</font></i>: 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">I</font></span><a href="zfmisc_1.html#K2">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">J</font></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">H</font> =  <a href="yellow20.html#K1" target="_self">Intersect</a> <font color="Maroon">F</font>,<font color="Maroon">G</font> as    <a href="pboole.html#M1">ManySortedSet</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">H</font>
;<br/>

<b>thus </b><a NAME="E3:20"/>
<font color="Maroon">H</font> <a href="hidden.html#R1">=</a>  <a href="yellow20.html#K1" target="_self">Intersect</a> <font color="Maroon">F</font>,<font color="Maroon">G</font>
 ;<br/>

<a NAME="E4:20"/><i><font color="Green">E32</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">I</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">J</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <font color="Maroon">H</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">F</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">I</font>,<font color="Maroon">I</font></span><a href="zfmisc_1.html#K3">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">G</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">J</font>,<font color="Maroon">J</font></span><a href="zfmisc_1.html#K3">:]</a></span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">H</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span> )
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:20"/>
( <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">I</font>,<font color="Maroon">I</font></span><a href="zfmisc_1.html#K3">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">I</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">I</font></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">J</font>,<font color="Maroon">J</font></span><a href="zfmisc_1.html#K3">:]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">J</font></span><a href="zfmisc_1.html#K2">:]</a></span>,<font color="Maroon">J</font></span><a href="zfmisc_1.html#K2">:]</a></span> )
 
<b>by </b><i><a class="ref" href="zfmisc_1.html#D3">ZFMISC_1:def 3</a></i>;<br/>
<b>then </b><i><font color="Green">E33</font></i>: <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">I</font>,<font color="Maroon">I</font></span><a href="zfmisc_1.html#K3">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">J</font>,<font color="Maroon">J</font></span><a href="zfmisc_1.html#K3">:]</a></span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>,<span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i>, <a class="ref" href="zfmisc_1.html#T123">ZFMISC_1:123</a></i>
<br/>.= 
<span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font></span>)</span></span><a href="zfmisc_1.html#K3">:]</a></span>
<b>by </b><i><a class="ref" href="zfmisc_1.html#D3">ZFMISC_1:def 3</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="20_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:20_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">I</font>,<font color="Maroon">I</font></span><a href="zfmisc_1.html#K3">:]</a></span> <a href="xboole_0.html#K3">/\</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">J</font>,<font color="Maroon">J</font></span><a href="zfmisc_1.html#K3">:]</a></span>
 ;<br/><a NAME="E2:20_2"/><b>then </b><i><font color="Green">E34</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">I</font>,<font color="Maroon">I</font>,<font color="Maroon">I</font></span><a href="zfmisc_1.html#K3">:]</a></span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K3">[:</a><span class="default"><font color="Maroon">J</font>,<font color="Maroon">J</font>,<font color="Maroon">J</font></span><a href="zfmisc_1.html#K3">:]</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">c</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:20_2"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="mcart_1.html#K3">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font></span><a href="mcart_1.html#K3">]</a></span> )
 <b>by </b><i><a class="ref" href="mcart_1.html#T72">MCART_1:72</a></i>;<br/><a NAME="E5:20_2"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">J</font> )
 <b>by </b><i>, <a class="ref" href="yellow20.html#T1" target="_self">Th1</a>, <a class="ref" href="mcart_1.html#T73">MCART_1:73</a></i>;<br/><a NAME="E6:20_2"/><b>then </b><i><font color="Green">E52</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font> )
 <b>by </b><i><a class="ref" href="yellow20.html#T1" target="_self">Th1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E7:20_2"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">I</font> <a href="xboole_0.html#K3">/\</a> <font color="Maroon">J</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T106">ZFMISC_1:106</a></i>;<br/><a NAME="E8:20_2"/><i><font color="Green">E54</font></i>: 
( <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">F</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">G</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> &amp; <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">H</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="multop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font> )
 <b>by </b><i><a class="ref" href="yellow20.html#T1" target="_self">Th1</a>, , , <a class="ref" href="altcat_1.html#D5">ALTCAT_1:def 5</a></i>;<br/><b>hence </b><span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">H</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> = 
<font color="Maroon">H</font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span>
<b>by </b><i><a class="ref" href="yellow20.html#T1" target="_self">Th1</a>, <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="funct_1.html#K1">.</a> <span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">c</font></span><a href="tarski.html#K4">]</a></span></span>)</span>
<b>by </b><i>, , , </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">F</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">a</font>,<font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow20.html#T1" target="_self">Th1</a>, , <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">F</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">G</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="yellow20.html#T1" target="_self">Th1</a>, , <a class="ref" href="multop_1.html#D1">MULTOP_1:def 1</a></i>
;<br/><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:20"/>
 <a href="yellow20.html#K1" target="_self">Intersect</a> <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">F</font></span><a href="altcat_1.html#K2">|}</a></span>,<span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">G</font></span><a href="altcat_1.html#K2">|}</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="altcat_1.html#K2">{|</a><span class="default"><font color="Maroon">H</font></span><a href="altcat_1.html#K2">|}</a></span>
 <b>by </b><i>, , </i>;<br/>


</div>
