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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">BL</font> be  non <a href="realset2.html#V3">trivial</a> <a href="lattices.html#NM8">B_Lattice</a>;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">BL</font>;<br/>





<a NAME="E1:18"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K8" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> implies  ex <font color="Olive">F</font> being   <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">BL</font> st <br/>( <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">F</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K8" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 ;<br/>

<a NAME="E2:18_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">UF</font> where <font color="Olive">UF</font> is    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">BL</font> : ( <font color="Olive">UF</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">UF</font> ) </span>}</span> 
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">F</font> being    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">BL</font><b> such that </b><br/><a NAME="E4:18_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">F</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 ;<br/>
<b>take </b>
<font color="Maroon">F</font>
;<br/>

<b>thus </b><a NAME="E5:18_1"/>
( <font color="Maroon">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">F</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> )
 <b>by </b><i><a class="ref" href="lopclset.html#T4" target="_self">Th4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:18"/>
(  ex <font color="Olive">F</font> being   <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">BL</font> st <br/>( <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">F</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> ) implies <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K8" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_2"/>
 ex <font color="Olive">F</font> being   <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">BL</font> st <br/>( <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">F</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> )
 ;<br/>

<a NAME="E2:18_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">UF</font> where <font color="Olive">UF</font> is    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">BL</font> : ( <font color="Olive">UF</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">UF</font> ) </span>}</span> 
 
;<br/>
<b>hence </b><a NAME="E3:18_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K8" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:18"/>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="lopclset.html#K8" target="_self">UFilter</a> <font color="Maroon">BL</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> iff  ex <font color="Olive">F</font> being   <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">BL</font> st <br/>( <font color="Olive">F</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Olive">F</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Olive">F</font> ) )
 <b>by </b><i><a class="ref" href="lopclset.html#T3" target="_self">Th3</a></i>;<br/>


</div>
