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<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">Z</font> be    <a href="pboole.html#M4">ManySortedSubset</a> of <font color="Maroon">M</font>;<br/>







<b>assume </b><a NAME="E1:38"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">Z1</font> being   <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">I</font>  st <font color="Olive">Z1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> holds <br/><font color="Maroon">Z</font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Olive">Z1</font>
 ;<br/>

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a><br/>

<b>assume </b><a NAME="E2:38"/><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>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="E4:38"/><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">SF</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="E5:38"/><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">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="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="E6:38"/>
<font color="Maroon">Z</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">M</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D23">PBOOLE:def 23</a></i>;<br/>
<a NAME="E7:38"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">Z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> is   <a href="subset_1.html#NM2">Subset</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>by </b><i>, <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">q</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:38_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Q</font>
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:38_1_1"/>
(  not <font color="Maroon">SF</font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">SF</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_1"><b>suppose </b></a><a NAME="E1:38_1_1_1"/>
not <font color="Maroon">SF</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:38_1_1_1"/><b>then </b>
<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="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"><font color="Olive">x</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> where <font color="Olive">x</font> is    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> </span>}</span> 
 <b>by </b><i>, </i>;<br/><b>then consider </b><font color="Maroon">a</font> being    <a href="closure2.html#M1" target="_self">Element</a> of  <a href="closure2.html#K2" target="_self">Bool</a> <font color="Maroon">M</font><b> such that </b><br/><a NAME="E4:38_1_1_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">SF</font> )
 <b>by </b><i>, </i>;<br/><a NAME="E5:38_1_1_1"/>
<font color="Maroon">Z</font> <a href="closure2.html#NR2" target="_self">c='</a> <font color="Maroon">a</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E6:38_1_1_1"/>
<font color="Maroon">Z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">q</font>
 <b>by </b><i>, , <a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="38_1_1_2"><b>suppose </b></a><a NAME="E1:38_1_1_2"/>
<font color="Maroon">SF</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><a NAME="E2:38_1_1_2"/><b>then </b>
<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>  <a href="pboole.html#K1">[0]</a> <font color="Maroon">I</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:38_1_1_2"/>
<font color="Maroon">Z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">q</font>
 <b>by </b><i>, , <a class="ref" href="pboole.html#T5">PBOOLE:5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:38"/>
<font color="Maroon">Z</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <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>
 <b>by </b><i>, , <a class="ref" href="mssubfam.html#T4">MSSUBFAM:4</a></i>;<br/>


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