<?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">SF</font> be   <a href="closure2.html#NM1">SubsetFamily</a> of <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">E</font> be    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font>;<br/><b>let </b><font color="Maroon">T</font> be    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font>;<br/>









<b>assume </b><a NAME="E1:37"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">SF</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">E</font>,<font color="Maroon">T</font></span><a href="tarski.html#K2">}</a></span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_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="E1:37_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><b>reconsider </b><font color="Maroon">sf1</font> = <font color="Maroon">SF</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="closure2.html#NM1">SubsetFamily</a> of <font color="Maroon">M</font> <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">Q</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="E4:37_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="closure2.html#K5" target="_self">|:</a><span class="default"><font color="Maroon">sf1</font></span><a href="closure2.html#K5" target="_self">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>and </b><br/><a NAME="E5:37_1"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <span class="p2"><a href="closure2.html#K5" target="_self">|:</a><span class="default"><font color="Maroon">sf1</font></span><a href="closure2.html#K5" target="_self">:|</a></span></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">Q</font>
 <b>by </b><i>, <a class="ref" href="mssubfam.html#D2">MSSUBFAM:def 2</a></i>;<br/><a NAME="E6:37_1"/>
<font color="Maroon">Q</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="pboole.html#D16">PBOOLE:def 16</a></i>;<br/><b>hence </b><span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <span class="p2"><a href="closure2.html#K5" target="_self">|:</a><span class="default"><font color="Maroon">SF</font></span><a href="closure2.html#K5" target="_self">:|</a></span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> = 
 <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><span class="p2"><a href="closure2.html#K5" target="_self">|:</a><span class="default"><font color="Maroon">sf1</font></span><a href="closure2.html#K5" target="_self">:|</a></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/>.= 
 <a href="setfam_1.html#K1">meet</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">E</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">T</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span><a href="tarski.html#K2">}</a></span>
<b>by </b><i>, , </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">E</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"><font color="Maroon">T</font> <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#T12">SETFAM_1:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">T</font></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>
<b>hence </b><a NAME="E3:37"/>
 <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5" target="_self">|:</a><span class="default"><font color="Maroon">SF</font></span><a href="closure2.html#K5" target="_self">:|</a></span> <a href="pboole.html#R6">=</a> <font color="Maroon">E</font> <a href="pboole.html#K3">/\</a> <font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
