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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="lattices.html#NM8">B_Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="filter_2.html#M2" target="_self">Ideal</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="filter_2.html#K15" target="_self">.:</a>  as    <a href="filter_2.html#M1" target="_self">Filter</a> of <font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a>  ;<br/>
<b>assume </b><a NAME="E2:84"/>
<font color="Maroon">c<sub>2</sub></font> <a href="filter_2.html#R2" target="_self">is_max-ideal</a> 
 ;<br/>

<a NAME="E3:84"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a> 
 
<b>by </b><i><a class="ref" href="filter_2.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E4:84"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a> </span>)</span> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> iff not <font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) ) )
 
<b>by </b><i><a class="ref" href="filter_0.html#T59">FILTER_0:59</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<a NAME="E5:84"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="filter_2.html#K5" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="filter_2.html#K5" target="_self">.:</a> </span>)</span> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K7">`</a>  )
 
<b>by </b><i><a class="txt" href="filter_2.html#E70"><i><font color="Green">Lemma30</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:84"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> iff not <font color="Maroon">c<sub>4</sub></font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="filter_2.html#E4:84"><i><font color="Green">E32</font></i></a></i>;<br/>


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