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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">Filter</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>







<b>assume </b><a NAME="E1:4"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">q</font></span><a href="filter_0.html#K2">.)</a></span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">F</font></span>)</span></span><a href="filter_0.html#K3">.)</a></span>
 ;<br/>

<a NAME="E2:4"/>
<span class="p1"><a href="filter_0.html#K3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">q</font></span><a href="filter_0.html#K2">.)</a></span> <a href="finsub_1.html#K1">\/</a> <font color="Maroon">F</font></span>)</span></span><a href="filter_0.html#K3">.)</a></span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> :  ex <font color="Olive">p'</font>, <font color="Olive">q'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">q'</font> <a href="lattices.html#R3">[=</a> <font color="Olive">r</font> &amp; <font color="Olive">p'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">q</font></span><a href="filter_0.html#K2">.)</a></span> &amp; <font color="Olive">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="filter_0.html#T48">FILTER_0:48</a></i>;<br/>
<a NAME="E3:4"/><b>then </b>
 ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  ex <font color="Olive">p'</font>, <font color="Olive">q'</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">q'</font> <a href="lattices.html#R3">[=</a> <font color="Olive">r</font> &amp; <font color="Olive">p'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">q</font></span><a href="filter_0.html#K2">.)</a></span> &amp; <font color="Olive">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> ) )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">p'</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <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="E5:4"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p'</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_0.html#K2">&lt;.</a><span class="default"><font color="Maroon">q</font></span><a href="filter_0.html#K2">.)</a></span> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>
<a NAME="E6:4"/>
<font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p'</font>
 
<b>by </b><i>, <a class="ref" href="filter_0.html#T18">FILTER_0:18</a></i>;<br/>
<a NAME="E7:4"/><b>then </b>
<font color="Maroon">q</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p'</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q'</font>
 
<b>by </b><i><a class="ref" href="lattices.html#T27">LATTICES:27</a></i>;<br/>
<a NAME="E8:4"/><b>then </b>
<font color="Maroon">q</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q'</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p</font>
 
<b>by </b><i>, <a class="ref" href="lattices.html#T25">LATTICES:25</a></i>;<br/>
<b>hence </b><a NAME="E9:4"/>
 ex <font color="Olive">r</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> &amp; <font color="Maroon">q</font> <a href="lattices.html#K4">"/\"</a> <font color="Olive">r</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p</font> )
 <b>by </b><i/>;<br/>


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