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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/>
<a NAME="E2:82"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="filter_0.html#NR1" target="_self">is_ultrafilter</a>  iff ( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a> ) &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> or <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#T57">FILTER_0:57</a></i>;<br/>
<b>thus </b><a NAME="E3:82"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="filter_2.html#R2" target="_self">is_max-ideal</a>  implies ( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or <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>2</sub></font> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:82_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="filter_2.html#R2" target="_self">is_max-ideal</a> 
 ;<br/>

<b>hence </b><a NAME="E2:82_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>)
 <b>by </b><i><a class="txt" href="filter_2.html#E2:82"><i><font color="Green">E32</font></i></a>, <a class="ref" href="filter_2.html#T33" target="_self">Th33</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/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <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> ;<br/>
<a NAME="E4:82_1"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or ( <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</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#E2:82"><i><font color="Green">E32</font></i></a>, <a class="txt" href="filter_2.html#E1:82_1"><i><font color="Green">E33</font></i></a>, <a class="ref" href="filter_2.html#T33" target="_self">Th33</a>, <a class="ref" href="filter_2.html#T55" target="_self">Th55</a></i>;<br/>
<b>hence </b><a NAME="E5:82_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or <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> )
 ;<br/>


</div><b>end;</b></div>

<b>assume </b><a NAME="E4:82"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>1</sub></font>) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> or <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>2</sub></font> ) ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <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>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><a NAME="E2:82_2"/>
( <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or ( <font color="Maroon">c<sub>5</sub></font> <a href="lattices.html#K7">`</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="filter_2.html#K6" target="_self">.:</a> <font color="Maroon">c<sub>4</sub></font></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#E4:82"><i><font color="Green">E33</font></i></a>, <a class="ref" href="filter_2.html#T55" target="_self">Th55</a></i>;<br/><b>hence </b><a NAME="E3:82_2"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> or <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>3</sub></font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:82"/>
<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="txt" href="filter_2.html#E2:82"><i><font color="Green">E32</font></i></a>, <a class="txt" href="filter_2.html#E4:82"><i><font color="Green">E33</font></i></a>, <a class="ref" href="filter_2.html#T33" target="_self">Th33</a></i>;<br/>


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