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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/>
<span class="p1"><a href="filter_2.html#K1" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a> </span>)</span></span><a href="filter_2.html#K1" target="_self">.)</a></span> = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a> )

<br/>.= 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)

<br/>.= 
<span class="p1"><a href="filter_2.html#K17" target="_self">(.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="filter_2.html#K17" target="_self">.&gt;</a></span>

;<br/>
<a NAME="E3:83"/><b>then </b>
( ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="filter_2.html#K17" target="_self">(.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="filter_2.html#K17" target="_self">.&gt;</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> is <a href="filter_0.html#V2">prime</a> ) iff <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_0.html#T58">FILTER_0:58</a></i>;<br/>
<b>hence </b><a NAME="E4:83"/>
( ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="filter_2.html#K17" target="_self">(.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="filter_2.html#K17" target="_self">.&gt;</a></span> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="filter_2.html#V1" target="_self">prime</a> ) iff <font color="Maroon">c<sub>2</sub></font> <a href="filter_2.html#R2" target="_self">is_max-ideal</a>  )
 <b>by </b><i><a class="ref" href="filter_2.html#T33" target="_self">Th33</a>, <a class="ref" href="filter_2.html#T44" target="_self">Th44</a></i>;<br/>


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