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<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM1" title="LATTICES:NM.1">Lattice</a>;<br/><b>let </b><font color="Maroon" title="c2">I</font> be    <a href="filter_2.html#M2" title="FILTER_2:mode.2">Ideal</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>thus </b><a NAME="E1:51"/>
( <font color="Maroon" title="c2">I</font> <a href="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a>  implies <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K15" title="FILTER_2:func.15">.:</a>  <a href="filter_0.html#NR1" title="FILTER_0:NR.1">is_ultrafilter</a>  )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c2">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L</font>) &amp; ( for <font color="Olive" title="b1">J</font> being   <a href="filter_2.html#M2" title="FILTER_2:mode.2">Ideal</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">J</font> &amp; <font color="Olive" title="b1">J</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L</font>) holds <br/><font color="Maroon" title="c2">I</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Olive" title="b1">J</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_2.html#D10" target="_self" title="FILTER_2:def.10">FILTER_2:def 10</a><br/>

<b>hence </b><a NAME="E2:51_1"/>
<font color="Maroon" title="c2">I</font> <a href="filter_2.html#K15" title="FILTER_2:func.15">.:</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D4" title="FILTER_0:def.4">FILTER_0:def 4</a><br/>

<b>let </b><font color="Maroon" title="c3">F</font> be    <a href="filter_2.html#M1" title="FILTER_2:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> ;<br/>

<a NAME="E3:51_1"/>
 <a href="filter_2.html#K14" title="FILTER_2:func.14">.:</a> <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font>
 
;<br/>
<b>hence </b><a NAME="E4:51_1"/>
(  not <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K15" title="FILTER_2:func.15">.:</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> or <font color="Maroon" title="c3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> or <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K15" title="FILTER_2:func.15">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F</font> )
 <b>by </b><i><a class="txt" href="filter_2.html#E1:51_1"><i><font color="Green" title="E20">A1</font></i></a></i>;<br/>


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

<b>assume </b><a NAME="E2:51"/><i><font color="Green" title="E20">A2</font></i>: 
( <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K15" title="FILTER_2:func.15">.:</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> ) &amp; ( for <font color="Olive" title="b1">F</font> being   <a href="filter_2.html#M1" title="FILTER_2:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>   st <font color="Maroon" title="c2">I</font> <a href="filter_2.html#K15" title="FILTER_2:func.15">.:</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">F</font> &amp; <font color="Olive" title="b1">F</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> ) holds <br/><font color="Maroon" title="c2">I</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="Olive" title="b1">F</font> ) )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_0.html#D4" title="FILTER_0:def.4">FILTER_0:def 4</a><br/>

<b>hence </b><a NAME="E3:51"/>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L</font>)
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_2.html#D10" target="_self" title="FILTER_2:def.10">FILTER_2:def 10</a><br/>

<b>let </b><font color="Maroon" title="c3">J</font> be    <a href="filter_2.html#M2" title="FILTER_2:mode.2">Ideal</a> of <font color="Maroon" title="c1">L</font>;<br/>

<a NAME="E4:51"/>
<font color="Maroon" title="c3">J</font> <a href="filter_2.html#K15" title="FILTER_2:func.15">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">J</font>
 
;<br/>
<b>hence </b><a NAME="E5:51"/>
( <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">J</font> &amp; <font color="Maroon" title="c3">J</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">L</font> implies <font color="Maroon" title="c2">I</font> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <font color="Maroon" title="c3">J</font> )
 <b>by </b><i><a class="txt" href="filter_2.html#E2:51"><i><font color="Green" title="E20">A2</font></i></a></i>;<br/>


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