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<b>let </b><font color="Maroon">I</font> be  non <a href="xboole_0.html#V1">empty</a>  <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">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="msualg_5.html#K5">EqRelLatt</a> <font color="Maroon">M</font></span>)</span>;<br/><b>let </b><font color="Maroon">X1</font> be   <a href="closure2.html#NM1">SubsetFamily</a> of <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="pboole.html#K10">|]</a></span>;<br/>







<b>assume </b><a NAME="E1:9"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">X1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">X</font>
 ;<br/>

<b>let </b><font color="Maroon">a</font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>, <font color="Maroon">b</font> be   <a href="msualg_5.html#NM1">Equivalence_Relation</a> of <font color="Maroon">M</font>;<br/>

<b>assume </b><a NAME="E2:9"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">a</font> <a href="pboole.html#R6">=</a>  <a href="mssubfam.html#K6">meet</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">X1</font></span><a href="closure2.html#K5">:|</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="9_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="E1:9_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 ;<br/><a NAME="E2:9_1"/><b>then </b><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">X1</font></span><a href="closure2.html#K5">:|</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">Element</a> of  <a href="closure2.html#K6">Bool</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="pboole.html#K10">|]</a></span> : <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X1</font> </span>}</span> 
 <b>by </b><i>, , <a class="ref" href="closure2.html#T15">CLOSURE2:15</a></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"><span class="p2"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="pboole.html#K10">|]</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span><b> such that </b><br/><a NAME="E4:9_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">X1</font></span><a href="closure2.html#K5">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <span class="p1">(<span class="default"><a href="mssubfam.html#K6">meet</a> <span class="p2"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">X1</font></span><a href="closure2.html#K5">:|</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/><b>reconsider </b><font color="Maroon">b'</font> = <font color="Maroon">b</font> as    <a href="closure2.html#M1">Element</a> of  <a href="closure2.html#K6">Bool</a> <span class="p1"><a href="pboole.html#K10">[|</a><span class="default"><font color="Maroon">M</font>,<font color="Maroon">M</font></span><a href="pboole.html#K10">|]</a></span> <b>by </b><i>, </i>;<br/><a NAME="E6:9_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">b'</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">X1</font></span><a href="closure2.html#K5">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, , </i>;<br/><a NAME="E7:9_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><span class="p2"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">X1</font></span><a href="closure2.html#K5">:|</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 NAME="E8:9_1"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="setfam_1.html#K1">meet</a> <span class="p1">(<span class="default"><span class="p2"><a href="closure2.html#K5">|:</a><span class="default"><font color="Maroon">X1</font></span><a href="closure2.html#K5">:|</a></span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span> holds <br/><font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="setfam_1.html#D1">SETFAM_1:def 1</a></i>;<br/><b>hence </b><a NAME="E9:9_1"/>
<font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 <b>by </b><i>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:9"/>
<font color="Maroon">a</font> <a href="pboole.html#R2">c=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="pboole.html#D5">PBOOLE:def 5</a></i>;<br/>


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