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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="Red">::  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_0.html#K3" title="FILTER_0:func.3">&lt;.</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_0.html#K3" title="FILTER_0:func.3">.)</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"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Olive" title="b1">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</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_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <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="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_0.html#K3" title="FILTER_0:func.3">.)</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="Olive" title="b1">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Olive" title="b2">D2</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</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="Red">::  thesis: </font></span></a><span class="hide"> ( <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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</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"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c2">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</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_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D2</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> )</span><br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">D1</font>, <font color="Maroon" title="c5">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="Red">::  thesis: </font></span></a><span class="hide"> ( <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="c4">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <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"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c4">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> )</span><br/><a NAME="E1:60_1"/>
( <font color="Maroon" title="c5">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font> &amp; <font color="Maroon" title="c4">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c4">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font></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_0.html#D5" target="_self" title="FILTER_0:def.5">Def5</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E2:60_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c5">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c4">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font></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="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/><a NAME="E3:60_1"/>
<font color="Maroon" title="c4">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D1</font></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_0.html#D5" target="_self" title="FILTER_0:def.5">Def5</a></span>;<br/><a NAME="E4:60_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></span>;<br/><span class="kw">hence </span><a NAME="E5:60_1"/>
<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="c4">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font></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_0.html#T27" target="_self" title="FILTER_0:th.27">Th27</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c4">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></span><br/><a NAME="E6:60_1"/>
<span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c4">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font></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_0.html#T27" target="_self" title="FILTER_0:th.27">Th27</a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E7:60_1"/><span class="kw">then </span>
<span class="p1"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c4">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font></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="txt" href="#E2:60_1"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><span class="kw">hence </span><a NAME="E8:60_1"/>
<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"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c4">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="c4">D1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c5">D2</font></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_0.html#D5" target="_self" title="FILTER_0:def.5">Def5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:60"/>
( <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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c2">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</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_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <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"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c2">D1</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</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_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D2</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p3"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><font color="Maroon" title="c3">D2</font></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">D2</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> )
 ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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