<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="lattices.html#NM1">Lattice</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:54"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V14">upper-bounded</a>
 ;<br/>

<a NAME="E2:54"/><b>then </b><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a>  is <a href="lattices.html#V13">lower-bounded</a> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="filter_2.html#K5" target="_self">.:</a>  &amp;  <a href="lattices.html#K5">Bottom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="lattice2.html#K1">.:</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="ref" href="lattice2.html#T64">LATTICE2:64</a>, <a class="ref" href="lattice2.html#T79">LATTICE2:79</a></i>;<br/>
<b>assume </b><a NAME="E3:54"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="lattices.html#K6">Top</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <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> <b> such that </b><br/><a NAME="E5:54"/><i><font color="Green">E21</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="filter_2.html#K5" target="_self">.:</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> &amp; <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="txt" href="filter_2.html#E2:54"><i><font color="Green">E20</font></i></a>, <a class="ref" href="filter_0.html#T24">FILTER_0:24</a></i>;<br/>
<b>take </b>
 <a href="filter_2.html#K14" target="_self">.:</a> <font color="Maroon">c<sub>3</sub></font>
;<br/>

<a NAME="E6:54"/>
(  <a href="filter_2.html#K14" target="_self">.:</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <span class="p1">(<span class="default"><a href="filter_2.html#K14" target="_self">.:</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="filter_2.html#K15" target="_self">.:</a>  <a href="hidden.html#R1">=</a>  <a href="filter_2.html#K14" target="_self">.:</a> <font color="Maroon">c<sub>3</sub></font> )
 
;<br/>
<b>hence </b><a NAME="E7:54"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="filter_2.html#K14" target="_self">.:</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="filter_2.html#K14" target="_self">.:</a> <font color="Maroon">c<sub>3</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#E5:54"><i><font color="Green">E21</font></i></a>, <a class="ref" href="filter_2.html#T33" target="_self">Th33</a></i>;<br/>


</div>
