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<b>reconsider </b><font color="Maroon">D</font> =  <a href="filter_2.html#K12" target="_self">.:</a> <font color="Maroon">I</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> ;<br/>
<a NAME="E2:36_1_1"/>
<font color="Maroon">D</font> is    <a href="filter_2.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_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="E1:36_1_1_1"/>
(  <a href="filter_2.html#K12" target="_self">.:</a> <font color="Maroon">I</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> ) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">L</font>) &amp; <font color="Maroon">p</font> <a href="filter_2.html#K5" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q</font> <a href="filter_2.html#K5" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> &amp; <font color="Maroon">p</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">q</font> &amp; ( for <font color="Olive">p'</font>, <font color="Olive">q'</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span> holds <br/> ( ( <font color="Olive">p'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Olive">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> ) iff <font color="Olive">p'</font> <a href="lattices.html#K3">"\/"</a> <font color="Olive">q'</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> ) ) )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:36_1_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="E3:36_1_1"/>
 <a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">I</font> is    <a href="filter_2.html#M1" target="_self">Filter</a> of <font color="Maroon">L</font>
 ;<br/>


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