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

<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"> for <font color="Olive" title="b1">D1</font>, <font color="Olive" title="b2">D2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">D2</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Olive" title="b1">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">D2</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> &amp; <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">D2</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Olive" title="b2">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> )</span><br/><b>let </b><font color="Maroon" title="c2">D1</font>, <font color="Maroon" title="c3">D2</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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"> ( <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c2">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> &amp; <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> )</span><br/>





<a NAME="E1:72"/>
( <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c2">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c2">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c2">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p4"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span></span>)</span> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K3" title="FILTER_2:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a> </span>)</span></span><a href="filter_2.html#K3" title="FILTER_2:func.3">.)</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span></span>)</span> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font> &amp; <font color="Maroon" title="c2">D1</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">D1</font> &amp; <font color="Maroon" title="c3">D2</font> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">D2</font> &amp; <span class="p1">(<span class="default"><span class="p2"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c2">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span> <a href="filter_2.html#K7" title="FILTER_2:func.7">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c2">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font> &amp; <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c2">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="filter_2.html#K15" title="FILTER_2:func.15">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c2">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> &amp; <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="filter_2.html#K15" title="FILTER_2:func.15">.:</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="filter_2.html#T37" target="_self" title="FILTER_2:th.37">Th37</a></i>;<br/>
<b>hence </b><a NAME="E2:72"/>
( <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c2">D1</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> &amp; <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_2.html#K19" title="FILTER_2:func.19">(.</a><span class="default"><font color="Maroon" title="c3">D2</font></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span></span>)</span></span><a href="filter_2.html#K19" title="FILTER_2:func.19">.&gt;</a></span> )
 <b>by </b><i><a class="ref" href="filter_0.html#T47" title="FILTER_0:th.47">FILTER_0:47</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>
