<?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">F1</font> be    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">F2</font> be    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">L</font>;<br/>





<b>consider </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><br/><a NAME="E2:1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font>
 <b>by </b><i><a class="ref" href="filter_0.html#T11">FILTER_0:11</a></i>;<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="E4:1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font>
 <b>by </b><i><a class="ref" href="filter_0.html#T11">FILTER_0:11</a></i>;<br/>
<a NAME="E5:1"/>
( <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">F1</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">F2</font> )
 
<b>by </b><i>, , <a class="ref" href="filter_0.html#T10">FILTER_0:10</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">D</font> = <font color="Maroon">F1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">F2</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E7:1"/>
<font color="Maroon">D</font> is    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D1">FILTER_0:def 1</a><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/>



<a NAME="E1:1_1"/>
( ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> implies ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> ) ) &amp; ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> implies ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> ) ) &amp; ( <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> implies ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> ) ) &amp; ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> implies ( <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> ) ) &amp; ( <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> implies <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">F2</font> ) &amp; ( <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">F2</font> implies ( <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F1</font> &amp; <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F2</font> ) ) )
 
<b>by </b><i><a class="ref" href="filter_0.html#D1">FILTER_0:def 1</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:1_1"/>
( (  not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> or  not <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> or <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> ) &amp; (  not <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> or ( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> &amp; <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> ) ) )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:1"/>
<font color="Maroon">F1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">F2</font> is    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">L</font>
 ;<br/>


</div>
