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<b>let </b><font color="Maroon" title="c1">B</font> be   <a href="lattices.html#NM8" title="LATTICES:NM.8">B_Lattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">b</font> holds <br/> ex <font color="Olive" title="b3">IB</font> being   <a href="filter_2.html#M2" title="FILTER_2:mode.2">Ideal</a> of <font color="Maroon" title="c1">B</font> st <br/>( <font color="Olive" title="b3">IB</font> <a href="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a>  &amp; ( ( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">IB</font> &amp; not <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">IB</font> ) or ( not <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">IB</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">IB</font> ) ) )</span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font> implies  ex <font color="Olive" title="b1">IB</font> being   <a href="filter_2.html#M2" title="FILTER_2:mode.2">Ideal</a> of <font color="Maroon" title="c1">B</font> st <br/>( <font color="Olive" title="b1">IB</font> <a href="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a>  &amp; ( ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">IB</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">IB</font> ) or ( not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">IB</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">IB</font> ) ) ) )</span><br/>





<b>assume </b><a NAME="E1:85"/>
<font color="Maroon" title="c2">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">IB</font> being   <a href="filter_2.html#M2" title="FILTER_2:mode.2">Ideal</a> of <font color="Maroon" title="c1">B</font> st <br/>( <font color="Olive" title="b1">IB</font> <a href="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a>  &amp; ( ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">IB</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">IB</font> ) or ( not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">IB</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">IB</font> ) ) )</span><br/>

<b>then consider </b><font color="Maroon" title="c4">FB</font> being    <a href="filter_2.html#M1" title="FILTER_2:mode.1">Filter</a> of <font color="Maroon" title="c1">B</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> <b> such that </b><br/><a NAME="E3:85"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c4">FB</font> <a href="filter_0.html#NR1" title="FILTER_0:NR.1">is_ultrafilter</a>  &amp; ( ( <font color="Maroon" title="c2">a</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">FB</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">FB</font> ) or ( not <font color="Maroon" title="c2">a</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">FB</font> &amp; <font color="Maroon" title="c3">b</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">FB</font> ) ) )
 <b>by </b><i><a class="ref" href="filter_0.html#T60" title="FILTER_0:th.60">FILTER_0:60</a></i>;<br/>
<b>take </b><font color="Maroon" title="c5">IB</font> =  <a href="filter_2.html#K14" title="FILTER_2:func.14">.:</a> <font color="Maroon" title="c4">FB</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">IB</font> <a href="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a>  &amp; ( ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">IB</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">IB</font> ) or ( not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">IB</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">IB</font> ) ) )</span><br/>

<a NAME="E4:85"/>
<font color="Maroon" title="c5">IB</font> <a href="filter_2.html#K15" title="FILTER_2:func.15">.:</a>  <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c4">FB</font>
 
;<br/>
<b>hence </b><a NAME="E5:85"/>
<font color="Maroon" title="c5">IB</font> <a href="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a> 
 <b>by </b><i><a class="txt" href="filter_2.html#E3:85"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="filter_2.html#T33" target="_self" title="FILTER_2:th.33">Th33</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">IB</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">IB</font> ) or ( not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">IB</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">IB</font> ) )</span><br/>

<b>thus </b><a NAME="E6:85"/>
( ( <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">IB</font> &amp; not <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">IB</font> ) or ( not <font color="Maroon" title="c2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">IB</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">IB</font> ) )
 <b>by </b><i><a class="txt" href="filter_2.html#E3:85"><i><font color="Green" title="E31">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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