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

<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>assume </b><a NAME="E1:74"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">L</font> is   <a href="filter_0.html#NM2">I_Lattice</a>
 ;<br/>

<b>let </b><font color="Maroon">p'</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>, <font color="Maroon">q'</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>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D7" target="_self">FILTER_0:def 7</a><br/>

<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">p'</font>, <font color="Maroon">q</font> = <font color="Maroon">q'</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#T10" target="_self">Th10</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E4:74"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font> &amp; ( for <font color="Olive">r1</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">r1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font> holds <br/><font color="Olive">r1</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r</font> ) )
 <b>by </b><i>, </i>;<br/>
<b>reconsider </b><font color="Maroon">I</font> = <font color="Maroon">L</font> as   <a href="filter_0.html#NM2">I_Lattice</a> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">i</font> = <font color="Maroon">p</font>, <font color="Maroon">j</font> = <font color="Maroon">q</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> ;<br/>
<b>reconsider </b><font color="Maroon">FI</font> = <font color="Maroon">F</font> as    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">I</font> ;<br/>
<a NAME="E8:74"/>
( <font color="Maroon">i</font> <a href="filter_0.html#K4" target="_self">=&gt;</a> <font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">i</font> <a href="filter_0.html#K4" target="_self">=&gt;</a> <font color="Maroon">j</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FI</font> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T18" target="_self">Th18</a>, <a class="ref" href="filter_0.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>then reconsider </b><font color="Maroon">r'</font> = <font color="Maroon">r</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#T10" target="_self">Th10</a></i>;<br/>
<b>take </b>
<font color="Maroon">r'</font>
;<br/>

<a NAME="E10:74"/>
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r'</font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T11" target="_self">Th11</a></i>;<br/>
<b>hence </b><a NAME="E11:74"/>
<font color="Maroon">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">r'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q'</font>
 <b>by </b><i><a class="ref" href="filter_0.html#T18" target="_self">Th18</a>, <a class="ref" href="filter_0.html#T12" target="_self">Th12</a></i>;<br/>

<b>let </b><font color="Maroon">s'</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>assume </b><a NAME="E12:74"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">s'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q'</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">s</font> = <font color="Maroon">s'</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#T10" target="_self">Th10</a></i>;<br/>
<a NAME="E14:74"/>
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">s'</font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T11" target="_self">Th11</a></i>;<br/>
<a NAME="E15:74"/><b>then </b>
<font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">s</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T19" target="_self">Th19</a>, <a class="ref" href="filter_0.html#T12" target="_self">Th12</a></i>;<br/>
<a NAME="E16:74"/><b>then </b>
<font color="Maroon">s</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">r</font>
 
<b>by </b><i><a class="ref" href="filter_0.html#T18" target="_self">Th18</a></i>;<br/>
<b>hence </b><a NAME="E17:74"/>
<font color="Maroon">s'</font> <a href="lattices.html#R1">[=</a> <font color="Maroon">r'</font>
 <b>by </b><i><a class="ref" href="filter_0.html#T12" target="_self">Th12</a></i>;<br/>


</div>
