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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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</a> implies for <font color="Olive" title="b1">I</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="Olive" title="b1">I</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> holds <br/> ex <font color="Olive" title="b2">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 <br/>( <font color="Olive" title="b1">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">J</font> &amp; <font color="Olive" title="b2">J</font> <a href="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a>  ) )</span><br/>

<b>assume </b><a NAME="E1:52"/>
<font color="Maroon" title="c1">L</font> is <a href="lattices.html#V14" title="LATTICES:attr.14">upper-bounded</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">I</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="Olive" title="b1">I</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> holds <br/> ex <font color="Olive" title="b2">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 <br/>( <font color="Olive" title="b1">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">J</font> &amp; <font color="Olive" title="b2">J</font> <a href="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a>  )</span><br/>

<a NAME="E2:52"/><b>then </b><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c1">L</font> <a href="lattice2.html#K1" title="LATTICE2:func.1">.:</a>  is <a href="lattices.html#V13" title="LATTICES:attr.13">lower-bounded</a> &amp; <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">L</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> ) )
 
<b>by </b><i><a class="ref" href="lattice2.html#T64" title="LATTICE2:th.64">LATTICE2:64</a></i>;<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">I</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  ex <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 <br/>( <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="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a>  ) )</span><br/>

<b>assume </b><a NAME="E3:52"/>
<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>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <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 <br/>( <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="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a>  )</span><br/>

<b>then consider </b><font color="Maroon" title="c3">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> <b> such that </b><br/><a NAME="E5:52"/><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="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> &amp; <font color="Maroon" title="c3">F</font> is <a href="filter_0.html#V1" title="FILTER_0:attr.1">being_ultrafilter</a> )
 <b>by </b><i><a class="txt" href="#E2:52"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="filter_0.html#T22" title="FILTER_0:th.22">FILTER_0:22</a></i>;<br/>
<b>take </b>
 <a href="filter_2.html#K14" title="FILTER_2:func.14">.:</a> <font color="Maroon" title="c3">F</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">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="filter_2.html#K14" title="FILTER_2:func.14">.:</a> <font color="Maroon" title="c3">F</font> &amp;  <a href="filter_2.html#K14" title="FILTER_2:func.14">.:</a> <font color="Maroon" title="c3">F</font> <a href="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a>  )</span><br/>

<a NAME="E6:52"/>
(  <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> &amp; <span class="p1">(<span class="default"><a href="filter_2.html#K14" title="FILTER_2:func.14">.:</a> <font color="Maroon" title="c3">F</font></span>)</span> <a href="filter_2.html#K15" title="FILTER_2:func.15">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="filter_2.html#K14" title="FILTER_2:func.14">.:</a> <font color="Maroon" title="c3">F</font> &amp; <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="c2">I</font> )
 
;<br/>
<b>hence </b><a NAME="E7:52"/>
( <font color="Maroon" title="c2">I</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="filter_2.html#K14" title="FILTER_2:func.14">.:</a> <font color="Maroon" title="c3">F</font> &amp;  <a href="filter_2.html#K14" title="FILTER_2:func.14">.:</a> <font color="Maroon" title="c3">F</font> <a href="filter_2.html#R2" title="FILTER_2:pred.2">is_max-ideal</a>  )
 <b>by </b><i><a class="txt" href="#E5:52"><i><font color="Green" title="E20">A2</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/>


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