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<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K7" title="FILTER_2:func.7">(.</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#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Olive" title="b1">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&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#K7" title="FILTER_2:func.7">.&gt;</a></span> &amp; <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</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#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</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#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Olive" title="b2">D2</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> )</span><br/><span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</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#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c2">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&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#K7" title="FILTER_2:func.7">.&gt;</a></span> &amp; <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</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#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</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#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c3">D2</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> )</span><br/>





<a NAME="E1:68"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
( <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</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#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c3">D2</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0: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#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c3">D2</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span></span>)</span> <a href="filter_2.html#K3" title="FILTER_2:func.3">.:</a></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c3">D2</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0: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#K3" title="FILTER_2:func.3">.:</a></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T36" target="_self" title="FILTER_2:th.36">Th37</a></span>;<br/>
<a NAME="E2:68"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
<span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</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#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0: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#K3" title="FILTER_2:func.3">.:</a></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T36" target="_self" title="FILTER_2:th.36">Th37</a></span>;<br/>
<a NAME="E3:68"/>
( <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c2">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&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#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0: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#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c2">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&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#K3" title="FILTER_2:func.3">.:</a></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c2">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0: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#K3" title="FILTER_2:func.3">.:</a></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T36" target="_self" title="FILTER_2:th.36">Th37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:68"/>
( <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</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#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c2">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&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#K7" title="FILTER_2:func.7">.&gt;</a></span> &amp; <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</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#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</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#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c3">D2</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span></span>)</span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:68"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:68"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="filter_0.html#T34" title="FILTER_0:th.34">FILTER_0:34</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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