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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="E1:85"/>
<font color="Maroon">a</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">b</font>
 ;<br/>

<b>then consider </b><font color="Maroon">FB</font> being    <a href="filter_2.html#M1" target="_self">Filter</a> of <font color="Maroon">B</font> <a href="lattice2.html#K1">.:</a> <b> such that </b><br/><a NAME="E3:85"/><i><font color="Green">E14</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="filter_2.html#K5" target="_self">.:</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> &amp; not <font color="Maroon">b</font> <a href="filter_2.html#K5" target="_self">.:</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> ) or ( not <font color="Maroon">a</font> <a href="filter_2.html#K5" target="_self">.:</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> &amp; <font color="Maroon">b</font> <a href="filter_2.html#K5" target="_self">.:</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">FB</font> ) ) )
 <b>by </b><i><a class="ref" href="filter_0.html#T60">FILTER_0:60</a></i>;<br/>
<b>take </b><font color="Maroon">IB</font> =  <a href="filter_2.html#K14" target="_self">.:</a> <font color="Maroon">FB</font>;<br/>

<a NAME="E4:85"/>
<font color="Maroon">IB</font> <a href="filter_2.html#K15" target="_self">.:</a>  <a href="filter_2.html#R1" target="_self">=</a> <font color="Maroon">FB</font>
 
;<br/>
<b>hence </b><a NAME="E5:85"/>
<font color="Maroon">IB</font> <a href="filter_2.html#R2" target="_self">is_max-ideal</a> 
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E6:85"/>
( ( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IB</font> &amp; not <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IB</font> ) or ( not <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IB</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">IB</font> ) )
 <b>by </b><i/>;<br/>


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