<?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">p</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Olive" title="b1">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</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" 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"> ( <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0: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#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> &amp; <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> )</span><br/>



<a NAME="E1:41"/>
<span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a>  <a href="filter_2.html#K4" title="FILTER_2:func.4">.:</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0: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#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_2.html#D1" target="_self" onmouseover="rs('filter_2/D1')" onmouseout="rh()">FILTER_2:def 1</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"> ( <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> iff <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="filter_2.html#K4" title="FILTER_2:func.4">.:</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0: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#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> )</span><br/>

<a NAME="E1:41_1"/><span class="lab"><font color="Green" title="E16">A1</font></span>: 
( <font color="Maroon" title="c3">q</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0: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#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> iff <font color="Maroon" title="c2">p</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a>  <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T15" onmouseover="rs('filter_0/T15')" onmouseout="rh()">FILTER_0:15</a></span>;<br/>
<a NAME="E2:41_1"/>
( <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> iff <font color="Maroon" title="c3">q</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">p</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T28" target="_self" onmouseover="rs('filter_2/T28')" onmouseout="rh()">Th28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:41_1"/>
( <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> iff <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="filter_2.html#K4" title="FILTER_2:func.4">.:</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0: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#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_1"><span class="lab"><font color="Green" title="E16">A1</font></span></a>, <a class="ref" href="lattice2.html#T38" onmouseover="rs('lattice2/T38')" onmouseout="rh()">LATTICE2:38</a>, <a class="ref" href="lattice2.html#T39" onmouseover="rs('lattice2/T39')" onmouseout="rh()">LATTICE2:39</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:41"/>
<span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0: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#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</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"> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span></span><br/>

<a NAME="E3:41"/>
 <a href="filter_2.html#K4" title="FILTER_2:func.4">.:</a> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> <a href="filter_2.html#R1" title="FILTER_2:pred.1">=</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="41_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">q</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="filter_2.html#D1" target="_self" onmouseover="rs('filter_2/D1')" onmouseout="rh()">FILTER_2:def 1</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"> ( <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="filter_2.html#K4" title="FILTER_2:func.4">.:</a> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> iff <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> )</span><br/>

<a NAME="E1:41_2"/><span class="lab"><font color="Green" title="E16">A2</font></span>: 
( <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> iff <font color="Maroon" title="c2">p</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T15" onmouseover="rs('filter_0/T15')" onmouseout="rh()">FILTER_0:15</a></span>;<br/>
<a NAME="E2:41_2"/>
( <font color="Maroon" title="c3">q</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a>  <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> iff <font color="Maroon" title="c3">q</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a>  <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">p</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_2.html#T28" target="_self" onmouseover="rs('filter_2/T28')" onmouseout="rh()">Th28</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:41_2"/>
( <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="filter_2.html#K4" title="FILTER_2:func.4">.:</a> <span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> iff <font color="Maroon" title="c3">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41_2"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="ref" href="lattice2.html#T38" onmouseover="rs('lattice2/T38')" onmouseout="rh()">LATTICE2:38</a>, <a class="ref" href="lattice2.html#T39" onmouseover="rs('lattice2/T39')" onmouseout="rh()">LATTICE2:39</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><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E4:41"/>
<span class="p1"><a href="filter_2.html#K6" title="FILTER_2:func.6">(.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="filter_2.html#K1" title="FILTER_2:func.1">.:</a></span>)</span></span><a href="filter_2.html#K6" title="FILTER_2:func.6">.&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="filter_0.html#K2" title="FILTER_0:func.2">&lt;.</a><span class="default"><font color="Maroon" title="c2">p</font></span><a href="filter_0.html#K2" title="FILTER_0:func.2">.)</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>
