<?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">H</font>, <font color="Olive" title="b2">F</font> being   <a href="filter_0.html#NM1" title="FILTER_0:NM.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="b2">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b1">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="b2">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Olive" title="b1">H</font></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">H</font>, <font color="Maroon" title="c3">F</font> be   <a href="filter_0.html#NM1" title="FILTER_0:NM.1">Filter</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_0.html#K3" title="FILTER_0:func.3">&lt;.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">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="c3">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c2">H</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></span><br/>





<a NAME="E1:51"/>
( <font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c2">H</font> &amp; <font color="Maroon" title="c2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c2">H</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T36" target="_self" onmouseover="rs('filter_0/T36')" onmouseout="rh()">Th36</a></span>;<br/>
<a NAME="E2:51"/><span class="kw">then </span>
<font color="Maroon" title="c3">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:51"/>
<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="c3">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">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="c3">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c2">H</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#T22" target="_self" onmouseover="rs('filter_0/T22')" onmouseout="rh()">Th22</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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"> <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="c3">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c2">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="c3">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">H</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></span><br/>

<a NAME="E4:51"/>
<font color="Maroon" title="c3">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c2">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="c3">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">H</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="51_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</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"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c2">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="c3">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">H</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:51_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c2">H</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="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="c3">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">H</font></span>)</span></span><a href="filter_0.html#K3" title="FILTER_0:func.3">.)</a></span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:51_1"/><span class="lab"><font color="Green" title="E34">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E4:51_1"/><span class="lab"><font color="Green" title="E35">A2</font></span>: 
<font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">F</font>
 <span class="kw">and </span><br/><a NAME="E5:51_1"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c2">H</font>
 ;<br/>
<a NAME="E6:51_1"/>
<font color="Maroon" title="c2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E7:51_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A4</font></span>: 
<font color="Maroon" title="c6">q</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:51_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a></span>;<br/>
<a NAME="E8:51_1"/><span class="lab"><font color="Green" title="E38">A5</font></span>: 
<font color="Maroon" title="c3">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">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="c3">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">H</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#D4" target="_self" onmouseover="rs('filter_0/D4')" onmouseout="rh()">Def4</a></span>;<br/>
<a NAME="E9:51_1"/>
<font color="Maroon" title="c3">F</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" onmouseover="rs('xboole_1/T7')" onmouseout="rh()">XBOOLE_1:7</a></span>;<br/>
<a NAME="E10:51_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">H</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:51_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:51_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="c3">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">H</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="#E3:51_1"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E7:51_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a>, <a class="txt" href="#E8:51_1"><span class="lab"><font color="Green" title="E38">A5</font></span></a>, <a class="ref" href="filter_0.html#T9" target="_self" onmouseover="rs('filter_0/T9')" onmouseout="rh()">Th9</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>
<a NAME="E5:51"/><span class="kw">then </span>
<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="c3">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c2">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="c3">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="filter_0.html#T22" target="_self" onmouseover="rs('filter_0/T22')" onmouseout="rh()">Th22</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:51"/>
<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="c3">F</font> <a href="filter_0.html#K5" title="FILTER_0:func.5">"/\"</a> <font color="Maroon" title="c2">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="c3">F</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c2">H</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#T21" target="_self" onmouseover="rs('filter_0/T21')" onmouseout="rh()">Th21</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>
