<?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">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/><b>let </b><font color="Maroon">D'</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="lattice2.html#K1">.:</a> </span>)</span>;<br/>





<a NAME="E1:57"/><i><font color="Green">E14</font></i>: 
( <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">D</font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span></span><a href="filter_2.html#K3" target="_self">.)</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">D</font></span><a href="filter_2.html#K3" target="_self">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">D'</font></span>)</span> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span></span><a href="filter_2.html#K3" target="_self">.)</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">D'</font></span>)</span></span><a href="filter_2.html#K3" target="_self">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">D'</font></span>)</span> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> <a href="filter_2.html#R1" target="_self">=</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">D'</font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> )
 
<b>by </b><i/>;<br/>
<a NAME="E2:57"/>
for <font color="Olive">L</font> being  <a href="lattices.html#NM1">Lattice</a><br/> for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Olive">L</font> holds  <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">D</font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span></span><a href="filter_2.html#K3" target="_self">.)</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Olive">D</font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><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">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<a NAME="E1:57_1"/>
( <font color="Maroon">D</font> <a href="filter_2.html#K7" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">D</font> &amp; <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">D</font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> <a href="filter_2.html#K15" target="_self">.:</a>  <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">D</font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> &amp;  <a href="filter_2.html#K14" target="_self">.:</a> <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span></span><a href="filter_2.html#K3" target="_self">.)</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span></span><a href="filter_2.html#K3" target="_self">.)</a></span> )
 
;<br/>
<a NAME="E2:57_1"/><b>then </b>
( <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">D</font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> &amp; ( <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span></span><a href="filter_2.html#K3" target="_self">.)</a></span> implies <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">D</font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span></span><a href="filter_2.html#K3" target="_self">.)</a></span> ) &amp; <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span></span><a href="filter_2.html#K3" target="_self">.)</a></span> &amp; ( <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">D</font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> implies <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span></span><a href="filter_2.html#K3" target="_self">.)</a></span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">D</font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> ) )
 
<b>by </b><i>, <a class="ref" href="filter_0.html#D5">FILTER_0:def 5</a></i>;<br/>
<b>hence </b><a NAME="E3:57_1"/>
<span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span></span><a href="filter_2.html#K3" target="_self">.)</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">D</font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:57"/>
( <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span></span><a href="filter_2.html#K3" target="_self">.)</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">D</font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> &amp; <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">D</font></span><a href="filter_2.html#K3" target="_self">.)</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">D</font> <a href="filter_2.html#K7" target="_self">.:</a> </span>)</span></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> &amp; <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><span class="p2">(<span class="default"><a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">D'</font></span>)</span></span><a href="filter_2.html#K3" target="_self">.)</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><font color="Maroon">D'</font></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> &amp; <span class="p1"><a href="filter_2.html#K3" target="_self">&lt;.</a><span class="default"><font color="Maroon">D'</font></span><a href="filter_2.html#K3" target="_self">.)</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="filter_2.html#K19" target="_self">(.</a><span class="default"><span class="p2">(<span class="default"><a href="filter_2.html#K8" target="_self">.:</a> <font color="Maroon">D'</font></span>)</span></span><a href="filter_2.html#K19" target="_self">.&gt;</a></span> )
 <b>by </b><i/>;<br/>


</div>
