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





<b>assume </b><a NAME="E1:64"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/>

<a NAME="E2:64"/><b>then </b>
(  not <font color="Maroon">a</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">b</font> or  not <font color="Maroon">b</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">a</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T26">LATTICES:26</a></i>;<br/>
<a NAME="E3:64"/><b>then </b>
( <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">B</font> or <font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">B</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E4:64"/><b>then </b>
(  ex <font color="Olive">FB</font> being   <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">B</font> st <br/>( <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">FB</font> &amp; <font color="Olive">FB</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  ) or  ex <font color="Olive">FB</font> being   <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">B</font> st <br/>( <font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">FB</font> &amp; <font color="Olive">FB</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  ) )
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">FB</font> being    <a href="filter_0.html#M1" target="_self">Filter</a> of <font color="Maroon">B</font><b> such that </b><br/><a NAME="E6:64"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">FB</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp; ( <font color="Maroon">a</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> or <font color="Maroon">b</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K7">`</a> </span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> ) )
 ;<br/>
<b>take </b>
<font color="Maroon">FB</font>
;<br/>

<b>thus </b><a NAME="E7:64"/>
<font color="Maroon">FB</font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a> 
 <b>by </b><i/>;<br/>

<a NAME="E8:64"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> &amp; <font color="Maroon">b</font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> ) or ( <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> &amp; <font color="Maroon">a</font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:64"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> ) or ( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> ) )
 <b>by </b><i>, </i>;<br/>


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