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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">p</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">D</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>  st <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">D</font> holds <br/><span class="p1"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="filter_2.html#K10" title="FILTER_2:func.10">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K11" title="FILTER_2:func.11">(.</a><span class="default"><font color="Olive" title="b2">D</font></span><a href="filter_2.html#K11" title="FILTER_2:func.11">.&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> for <font color="Olive" title="b1">D</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>  st <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">D</font> holds <br/><span class="p1"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K10" title="FILTER_2:func.10">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K11" title="FILTER_2:func.11">(.</a><span class="default"><font color="Olive" title="b1">D</font></span><a href="filter_2.html#K11" title="FILTER_2:func.11">.&gt;</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">D</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"> ( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> implies <span class="p1"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K10" title="FILTER_2:func.10">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K11" title="FILTER_2:func.11">(.</a><span class="default"><font color="Maroon" title="c3">D</font></span><a href="filter_2.html#K11" title="FILTER_2:func.11">.&gt;</a></span> )</span><br/>





<a NAME="E1:60"/>
( <span class="p1"><a href="filter_2.html#K2" title="FILTER_2:func.2">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K5" title="FILTER_2:func.5">.:</a> </span>)</span></span><a href="filter_2.html#K2" title="FILTER_2:func.2">.)</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K10" title="FILTER_2:func.10">.&gt;</a></span> &amp; <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">D</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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_2.html#K11" title="FILTER_2:func.11">(.</a><span class="default"><font color="Maroon" title="c3">D</font></span><a href="filter_2.html#K11" title="FILTER_2:func.11">.&gt;</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T30" target="_self" title="FILTER_2:th.30">Th30</a>, <a class="ref" href="filter_2.html#T37" target="_self" title="FILTER_2:th.37">Th37</a></span>;<br/>
<span class="kw">hence </span><a NAME="E2:60"/>
( <font color="Maroon" title="c2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">D</font> implies <span class="p1"><a href="filter_2.html#K10" title="FILTER_2:func.10">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K10" title="FILTER_2:func.10">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K11" title="FILTER_2:func.11">(.</a><span class="default"><font color="Maroon" title="c3">D</font></span><a href="filter_2.html#K11" title="FILTER_2:func.11">.&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T29" title="FILTER_0:th.29">FILTER_0:29</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/>


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