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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="realset2.html#V3">trivial</a> <a href="lattices.html#NM8">B_Lattice</a>;<br/>

<b>assume </b><a NAME="E1:38"/><i><font color="Green">E33</font></i>: 
<span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><a href="lopclset.html#K9" target="_self">UFilter</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E3:38"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="filter_0.html#M1">Filter</a> of <font color="Maroon">c<sub>1</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  &amp;  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="lopclset.html#E1:38"><i><font color="Green">E33</font></i></a>, <a class="ref" href="lopclset.html#T20" target="_self">Th20</a></i>;<br/>
<b>hence </b><a NAME="E4:38"/>
contradiction
 <b>by </b><i><a class="ref" href="lopclset.html#T35" target="_self">Th35</a></i>;<br/>


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