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

<b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">H</font> being   <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font> holds  <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">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">H</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">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Olive" title="b2">H</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></span><br/><b>let </b><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">H</font> be    <a href="filter_0.html#M1" title="FILTER_0:mode.1">Filter</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</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">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">H</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></span><br/>





<a NAME="E1:53"/>
( <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Maroon" title="c3">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">H</font> )
 
<b>by </b><i><a class="ref" href="filter_0.html#T49" target="_self" title="FILTER_0:th.49">Th49</a></i>;<br/>
<a NAME="E2:53"/><b>then </b>
<font color="Maroon" title="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">H</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></i>;<br/>
<b>hence </b><a NAME="E3:53"/>
<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">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</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">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">H</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
 <b>by </b><i><a class="ref" href="filter_0.html#T27" target="_self" title="FILTER_0:th.27">Th27</a></i>; <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></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">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">H</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">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></span><br/>

<a NAME="E4:53"/>
<font color="Maroon" title="c2">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">H</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="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">H</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> )</span><br/>

<b>assume </b><a NAME="E1:53_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">H</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></span><br/>

<b>then consider </b><font color="Maroon" title="c5">p</font>, <font color="Maroon" title="c6">q</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:53_1"/><i><font color="Green" title="E39">A1</font></i>: 
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">p</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c6">q</font> &amp; <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">H</font> )
 ;<br/>
<a NAME="E4:53_1"/>
( <font color="Maroon" title="c2">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Maroon" title="c3">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</font> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E5:53_1"/><b>then </b>
( <font color="Maroon" title="c5">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Maroon" title="c6">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</font> &amp; <font color="Maroon" title="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</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="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> &amp; <font color="Maroon" title="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</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="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> )
 
<b>by </b><i><a class="txt" href="#E3:53_1"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="filter_0.html#D5" target="_self" title="FILTER_0:def.5">Def5</a></i>;<br/>
<b>hence </b><a NAME="E6:53_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
 <b>by </b><i><a class="txt" href="#E3:53_1"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="filter_0.html#T9" target="_self" title="FILTER_0:th.9">Th9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E5:53"/><b>then </b>
<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">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">H</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"><a href="filter_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
 
<b>by </b><i><a class="ref" href="filter_0.html#T27" target="_self" title="FILTER_0:th.27">Th27</a></i>;<br/>
<b>hence </b><a NAME="E6:53"/>
<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">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c3">H</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">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c3">H</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
 <b>by </b><i><a class="ref" href="filter_0.html#T26" target="_self" title="FILTER_0:th.26">Th26</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
