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

<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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">D</font>, <font color="Olive" title="b2">D1</font>, <font color="Olive" title="b3">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/> ( ( <font color="Olive" title="b2">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">D2</font> implies <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Olive" title="b2">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Olive" title="b3">D2</font></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"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Olive" title="b1">D</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Olive" title="b1">D</font></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">D</font>, <font color="Maroon" title="c3">D1</font>, <font color="Maroon" title="c4">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c3">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D2</font> implies <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></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"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c2">D</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c2">D</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> )</span><br/>







<a NAME="E1:52"/>
<font color="Maroon" title="c4">D2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></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="ref" href="filter_2.html#D9" target="_self" onmouseover="rs('filter_2/D9')" onmouseout="rh()">Def9</a></span>;<br/>
<a NAME="E2:52"/><span class="kw">then </span>
( <font color="Maroon" title="c3">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D2</font> implies <font color="Maroon" title="c3">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> )
 
;<br/>
<span class="kw">hence </span><a NAME="E3:52"/>
( ( <font color="Maroon" title="c3">D1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">D2</font> implies <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c3">D1</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c4">D2</font></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"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c2">D</font></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span></span><a href="filter_2.html#K7" title="FILTER_2:func.7">.&gt;</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="filter_2.html#K7" title="FILTER_2:func.7">(.</a><span class="default"><font color="Maroon" title="c2">D</font></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="ref" href="filter_2.html#D9" target="_self" onmouseover="rs('filter_2/D9')" onmouseout="rh()">Def9</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
