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<b>let </b><font color="Maroon">L</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">F</font> be    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font>;<br/>



<b>given </b><font color="Maroon">p</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><a NAME="E1:71"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( <font color="Maroon">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Olive">q</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14">LATTICES:def 14</a><br/>

<b>consider </b><font color="Maroon">q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:71"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font>
 <b>by </b><i/>;<br/>
<a NAME="E4:71"/><i><font color="Green">E43</font></i>: 
 ex <font color="Olive">o1</font>, <font color="Olive">o2</font> being  <a href="binop_1.html#NM2">BinOp</a> of <font color="Maroon">F</font> st <br/>( <font color="Olive">o1</font> <a href="hidden.html#R1">=</a> the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">L</font> <a href="realset1.html#K1">||</a> <font color="Maroon">F</font> &amp; <font color="Olive">o2</font> <a href="hidden.html#R1">=</a> the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">L</font> <a href="realset1.html#K1">||</a> <font color="Maroon">F</font> &amp;  <a href="filter_0.html#K8" target="_self">latt</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# <font color="Maroon">F</font>,<font color="Olive">o1</font>,<font color="Olive">o2</font> #) )
 
<b>by </b><i/>;<br/>
<a NAME="E5:71"/>
( <font color="Maroon">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T15" target="_self">Th15</a>, , </i>;<br/>
<b>then reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">p</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K8" target="_self">latt</a> <font color="Maroon">F</font></span>)</span> <b>by </b><i><a class="ref" href="filter_0.html#T18" target="_self">Th18</a></i>;<br/>
<b>take </b>
<font color="Maroon">p'</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D14">LATTICES:def 14</a><br/>

<b>let </b><font color="Maroon">r</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="filter_0.html#K8" target="_self">latt</a> <font color="Maroon">F</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">r'</font> = <font color="Maroon">r</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">F</font> <b>by </b><i><a class="ref" href="filter_0.html#T18" target="_self">Th18</a></i>;<br/>
<b>thus </b><font color="Maroon">p'</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">r</font> = 
<font color="Maroon">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">r'</font>
<b>by </b><i><a class="ref" href="filter_0.html#T11" target="_self">Th11</a></i>
<br/>.= 
<font color="Maroon">p'</font>
<b>by </b><i><a class="ref" href="filter_0.html#T15" target="_self">Th15</a></i>
;<br/>

<b>hence </b><a NAME="E9:71"/>
<font color="Maroon">r</font> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font>
 ;<br/>


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