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

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



<b>assume </b><a NAME="E1:92"/><i><font color="Green">E14</font></i>: 
<font color="Maroon">L</font> is <a href="lattices.html#V13">lower-bounded</a>
 ;<br/>

<b>let </b><font color="Maroon">q</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="filter_2.html#D1" target="_self">FILTER_2:def 1</a><br/>

<a NAME="E2:92"/>
(  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p</font> &amp;  <a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">q</font> )
 
<b>by </b><i>, <a class="ref" href="lattices.html#T41">LATTICES:41</a></i>;<br/>
<a NAME="E3:92"/><b>then </b>
( ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_2.html#K18" target="_self">(.</a><span class="default"><font color="Maroon">p</font></span><a href="filter_2.html#K18" target="_self">.&gt;</a></span> implies <font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p</font> ) &amp; ( <font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p</font> implies <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_2.html#K18" target="_self">(.</a><span class="default"><font color="Maroon">p</font></span><a href="filter_2.html#K18" target="_self">.&gt;</a></span> ) &amp; ( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">p</font></span><a href="filter_2.html#K22" target="_self">#]</a></span> implies <font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p</font> ) &amp; ( <font color="Maroon">q</font> <a href="lattices.html#R3">[=</a> <font color="Maroon">p</font> implies <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">p</font></span><a href="filter_2.html#K22" target="_self">#]</a></span> ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:92"/>
( <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_2.html#K18" target="_self">(.</a><span class="default"><font color="Maroon">p</font></span><a href="filter_2.html#K18" target="_self">.&gt;</a></span> iff <font color="Maroon">q</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="filter_2.html#K22" target="_self">[#</a><span class="default"><span class="p2">(<span class="default"><a href="lattices.html#K5">Bottom</a> <font color="Maroon">L</font></span>)</span>,<font color="Maroon">p</font></span><a href="filter_2.html#K22" target="_self">#]</a></span> )
 ;<br/>


</div>
