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

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





<a NAME="E1:22"/><i><font color="Green">E25</font></i>: 
<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> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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">b</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>set </b><font color="Maroon">c</font> = <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font>;<br/>
<b>assume </b><a NAME="E1:22_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> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">F0</font> being    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">BL</font><b> such that </b><br/><a NAME="E3:22_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F0</font> &amp; <font color="Maroon">F0</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F0</font> )
 <b>by </b><i><a class="ref" href="lopclset.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E4:22_1"/>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F0</font> or <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F0</font> )
 
<b>by </b><i><a class="txt" href="lopclset.html#E1:22"><i><font color="Green">E25</font></i></a>, <a class="ref" href="lopclset.html#T4" target="_self">Th4</a></i>;<br/>
<a NAME="E5:22_1"/><b>then </b>
( <font color="Maroon">F0</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> or <font color="Maroon">F0</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">b</font> )
 
<b>by </b><i><a class="txt" href="lopclset.html#E1:22"><i><font color="Green">E25</font></i></a>, <a class="ref" href="lopclset.html#T2" target="_self">Th2</a></i>;<br/>
<b>hence </b><a NAME="E6:22_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="lopclset.html#E1:22"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:22"/>
<span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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">b</font></span>)</span> <a href="tarski.html#R1">c=</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> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:22_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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">b</font></span>)</span>
 ;<br/>

<a NAME="E2:22_2"/><b>then </b>
( <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> or <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">b</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:22_2"/><b>then </b>
(  ex <font color="Olive">F0</font> being   <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">BL</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">F0</font> &amp; <font color="Olive">F0</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">F0</font> ) or  ex <font color="Olive">F0</font> being   <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">BL</font> st <br/>( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">F0</font> &amp; <font color="Olive">F0</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Olive">F0</font> ) )
 
<b>by </b><i><a class="ref" href="lopclset.html#T2" target="_self">Th2</a></i>;<br/>
<b>then consider </b><font color="Maroon">F0</font> being    <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">BL</font><b> such that </b><br/><a NAME="E5:22_2"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F0</font> &amp; <font color="Maroon">F0</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">F0</font> or <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F0</font> ) )
 ;<br/>
<a NAME="E6:22_2"/>
( <font color="Maroon">F0</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; <font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F0</font> )
 
<b>by </b><i><a class="txt" href="lopclset.html#E1:22"><i><font color="Green">E25</font></i></a>, <a class="ref" href="lopclset.html#T4" target="_self">Th4</a></i>;<br/>
<b>hence </b><a NAME="E7:22_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> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="lopclset.html#E1:22"><i><font color="Green">E25</font></i></a>, <a class="ref" href="lopclset.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:22"/>
<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> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<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></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<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">b</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
