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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">IB</font> being   <a href="filter_2.html#M2" title="FILTER_2:mode.2">Ideal</a> of <font color="Maroon" title="c1">B</font> holds <br/> ( <font color="Olive" title="b1">IB</font> <a href="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a>  iff ( <font color="Olive" title="b1">IB</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">B</font> &amp; ( for <font color="Olive" title="b2">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font>  holds <br/>( <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">IB</font> or <font color="Olive" title="b2">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">IB</font> ) ) ) )</span><br/><b>let </b><font color="Maroon" title="c2">IB</font> be    <a href="filter_2.html#M2" title="FILTER_2:mode.2">Ideal</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">IB</font> <a href="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a>  iff ( <font color="Maroon" title="c2">IB</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">B</font> &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font>  holds <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IB</font> or <font color="Olive" title="b1">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IB</font> ) ) ) )</span><br/>



<b>reconsider </b><font color="Maroon" title="c3">FB</font> = <font color="Maroon" title="c2">IB</font> <a href="filter_2.html#K15" title="FILTER_2:func.15">.:</a>  as    <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>  ;<br/>
<a NAME="E2:82"/><i><font color="Green" title="E31">A1</font></i>: 
( <font color="Maroon" title="c3">FB</font> <a href="filter_0.html#NR1" title="FILTER_0:NR.1">is_ultrafilter</a>  iff ( <font color="Maroon" title="c3">FB</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">B</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> ) &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">B</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>  holds <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FB</font> or <font color="Olive" title="b1">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FB</font> ) ) ) )
 
<b>by </b><i><a class="ref" href="filter_0.html#T57" title="FILTER_0:th.57">FILTER_0:57</a></i>;<br/>
<b>thus </b><a NAME="E3:82"/>
( <font color="Maroon" title="c2">IB</font> <a href="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a>  implies ( <font color="Maroon" title="c2">IB</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">B</font>) &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font>  holds <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IB</font> or <font color="Olive" title="b1">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IB</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">IB</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">B</font> &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font>  holds <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IB</font> or <font color="Olive" title="b1">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IB</font> ) ) implies <font color="Maroon" title="c2">IB</font> <a href="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a>  )</span><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" title="E32">A2</font></i>: 
<font color="Maroon" title="c2">IB</font> <a href="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">IB</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">B</font>) &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font>  holds <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IB</font> or <font color="Olive" title="b1">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IB</font> ) ) )</span><br/>

<b>hence </b><a NAME="E2:82_1"/>
<font color="Maroon" title="c2">IB</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">B</font>)
 <b>by </b><i><a class="txt" href="filter_2.html#E2:82"><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"> for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font>  holds <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IB</font> or <font color="Olive" title="b1">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IB</font> )</span><br/>

<b>let </b><font color="Maroon" title="c4">a</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="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IB</font> or <font color="Maroon" title="c4">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IB</font> )</span><br/>

<b>reconsider </b><font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c4">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">B</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span> ;<br/>
<a NAME="E4:82_1"/>
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FB</font> or ( <font color="Maroon" title="c5">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FB</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">a</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  ) )
 
<b>by </b><i><a class="txt" href="filter_2.html#E2:82"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="filter_2.html#E1:82_1"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="filter_2.html#T33" target="_self" title="FILTER_2:th.33">Th33</a>, <a class="ref" href="filter_2.html#T55" target="_self" title="FILTER_2:th.55">Th55</a></i>;<br/>
<b>hence </b><a NAME="E5:82_1"/>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IB</font> or <font color="Maroon" title="c4">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IB</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E4:82"/><i><font color="Green" title="E32">A3</font></i>: 
( <font color="Maroon" title="c2">IB</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">B</font>) &amp; ( for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font>  holds <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IB</font> or <font color="Olive" title="b1">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">IB</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">IB</font> <a href="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a> </span><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" title="c4">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c1">B</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a> </span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FB</font> or <font color="Maroon" title="c4">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FB</font> )</span><br/><b>reconsider </b><font color="Maroon" title="c5">b</font> = <font color="Maroon" title="c4">a</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">B</font> ;<br/><a NAME="E2:82_2"/>
( <font color="Maroon" title="c5">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FB</font> or ( <font color="Maroon" title="c5">b</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FB</font> &amp; <span class="p1">(<span class="default"><a href="filter_2.html#K6" title="FILTER_2:func.6">.:</a> <font color="Maroon" title="c4">a</font></span>)</span> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  ) )
 <b>by </b><i><a class="txt" href="filter_2.html#E4:82"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="filter_2.html#T55" target="_self" title="FILTER_2:th.55">Th55</a></i>;<br/><b>hence </b><a NAME="E3:82_2"/>
( <font color="Maroon" title="c4">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FB</font> or <font color="Maroon" title="c4">a</font> <a href="lattices.html#K7" title="LATTICES:func.7">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">FB</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E6:82"/>
<font color="Maroon" title="c2">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#E2:82"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="filter_2.html#E4:82"><i><font color="Green" title="E32">A3</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"> verum</span><br/>


</div>
