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

<b>let </b><font color="Maroon" title="c1">BL</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="lattices.html#NM8" title="LATTICES:NM.8">B_Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font><br/> for <font color="Olive" title="b2">F</font> being   <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">BL</font> holds <br/> ( <font color="Olive" title="b2">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K8" title="LOPCLSET:func.8">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">a</font> iff ( <font color="Olive" title="b2">F</font> <a href="filter_0.html#NR1" title="FILTER_0:NR.1">is_ultrafilter</a>  &amp; <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">F</font> ) )</span><br/><b>let </b><font color="Maroon" title="c2">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">BL</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font> being   <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">BL</font> holds <br/> ( <font color="Olive" title="b1">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K8" title="LOPCLSET:func.8">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">a</font> iff ( <font color="Olive" title="b1">F</font> <a href="filter_0.html#NR1" title="FILTER_0:NR.1">is_ultrafilter</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> ) )</span><br/><b>let </b><font color="Maroon" title="c3">F</font> be    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">BL</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K8" title="LOPCLSET:func.8">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">a</font> iff ( <font color="Maroon" title="c3">F</font> <a href="filter_0.html#NR1" title="FILTER_0:NR.1">is_ultrafilter</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> ) )</span><br/>





<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> <a href="filter_0.html#NR1" title="FILTER_0:NR.1">is_ultrafilter</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> implies <font color="Maroon" title="c3">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K8" title="LOPCLSET:func.8">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">a</font> )</span>
<div class="add"><b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon" title="c3">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K8" title="LOPCLSET:func.8">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">a</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">F</font> <a href="filter_0.html#NR1" title="FILTER_0:NR.1">is_ultrafilter</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> )</span><br/><b>then consider </b><font color="Maroon" title="c4">F0</font> being    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">BL</font><b> such that </b><br/><a NAME="E3:18_1"/><i><font color="Green" title="E15">A1</font></i>: 
( <font color="Maroon" title="c4">F0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c4">F0</font> <a href="filter_0.html#NR1" title="FILTER_0:NR.1">is_ultrafilter</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">F0</font> )
 <b>by </b><i><a class="ref" href="lopclset.html#T20" target="_self" title="LOPCLSET:th.20">Th20</a></i>;<br/><b>thus </b><a NAME="E4:18_1"/>
( <font color="Maroon" title="c3">F</font> <a href="filter_0.html#NR1" title="FILTER_0:NR.1">is_ultrafilter</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> )
 <b>by </b><i><a class="txt" href="lopclset.html#E3:18_1"><i><font color="Green" title="E15">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:18"/>
( <font color="Maroon" title="c3">F</font> <a href="filter_0.html#NR1" title="FILTER_0:NR.1">is_ultrafilter</a>  &amp; <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> implies <font color="Maroon" title="c3">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K8" title="LOPCLSET:func.8">UFilter</a> <font color="Maroon" title="c1">BL</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c2">a</font> )
 <b>by </b><i><a class="ref" href="lopclset.html#T20" target="_self" title="LOPCLSET:th.20">Th20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
