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

<b>let </b><font color="Maroon">I</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">M</font> be    <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>;<br/><b>let </b><font color="Maroon">SH</font> be   <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">SF</font> be   <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">SG</font> be   <a href="mssubfam.html#NM1">MSSubsetFamily</a> of <font color="Maroon">M</font>;<br/>









<b>assume </b><a NAME="E1:72"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">SH</font> <a href="pboole.html#R6">=</a> <font color="Maroon">SF</font> <a href="pboole.html#K2">\/</a> <font color="Maroon">SG</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:72_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>then consider </b><font color="Maroon">Qf</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span><b> such that </b><br/><a NAME="E3:72_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">Qf</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SF</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E4:72_1"/><i><font color="Green">E44</font></i>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Qf</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">Qg</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span><b> such that </b><br/><a NAME="E6:72_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">Qg</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SG</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E7:72_1"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Qg</font>
 <b>by </b><i>, </i>;<br/><b>consider </b><font color="Maroon">Qh</font> being   <a href="setfam_1.html#NM1">Subset-Family</a> of <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span><b> such that </b><br/><a NAME="E9:72_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">Qh</font> <a href="hidden.html#R1">=</a> <font color="Maroon">SH</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E10:72_1"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SH</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Qh</font>
 <b>by </b><i>, </i>;<br/><a NAME="E11:72_1"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">Qh</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Qf</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">Qg</font>
 <b>by </b><i>, , , , , <a class="ref" href="pboole.html#D7">PBOOLE:def 7</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="72_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:72_1_1_1"/>
( ( <font color="Maroon">Qf</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Qg</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">Qf</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Qg</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">Qf</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Qg</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  ) or ( <font color="Maroon">Qf</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Qg</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_1_1"><b>case </b></a><a NAME="E1:72_1_1_1_1"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">Qf</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Qg</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><a NAME="E2:72_1_1_1_1"/><b>then </b>
<font color="Maroon">Qh</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SH</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="setfam_1.html#K6">meet</a> <font color="Maroon">Qh</font>
<b>by </b><i>, <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">Qf</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">Qg</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="setfam_1.html#T10">SETFAM_1:10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K6">meet</a> <font color="Maroon">Qg</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SF</font></span>)</span> <a href="pboole.html#K3">/\</a> <span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SG</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_1_2"><b>case </b></a><a NAME="E1:72_1_1_1_2"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">Qf</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Qg</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SH</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SF</font></span>)</span> <a href="pboole.html#K3">/\</a> <span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SG</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_1_3"><b>case </b></a><a NAME="E1:72_1_1_1_3"/><i><font color="Green">E78</font></i>: 
( <font color="Maroon">Qf</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Qg</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SH</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , , <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SF</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SG</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="setfam_1.html#D10">SETFAM_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SF</font></span>)</span> <a href="pboole.html#K3">/\</a> <span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SG</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>
;<br/><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="72_1_1_1_4"><b>case </b></a><a NAME="E1:72_1_1_1_4"/>
( <font color="Maroon">Qf</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">Qg</font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 ;<br/><div class="add"><b>hence </b><span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SH</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Qf</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="setfam_1.html#K8">Intersect</a> <font color="Maroon">Qg</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SF</font></span>)</span> <a href="pboole.html#K3">/\</a> <span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SG</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, , , <a class="ref" href="pboole.html#D8">PBOOLE:def 8</a></i>
;<br/><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E13:72_1"/>
<span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SH</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SF</font></span>)</span> <a href="pboole.html#K3">/\</a> <span class="p2">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SG</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:72"/>
 <a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SH</font> <a href="pboole.html#R6">=</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SF</font></span>)</span> <a href="pboole.html#K3">/\</a> <span class="p1">(<span class="default"><a href="mssubfam.html#K6" target="_self">meet</a> <font color="Maroon">SG</font></span>)</span>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
