<?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">X</font>, <font color="Olive" title="b2">Y</font>, <font color="Olive" title="b3">Z</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/> ( <font color="Olive" title="b1">X</font> <a href="boolealg.html#R1" title="BOOLEALG:pred.1">=</a> <font color="Olive" title="b2">Y</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b3">Z</font> iff ( <font color="Olive" title="b2">Y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">X</font> &amp; <font color="Olive" title="b3">Z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">X</font> &amp; ( for <font color="Olive" title="b4">V</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">Y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b4">V</font> &amp; <font color="Olive" title="b3">Z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b4">V</font> holds <br/><font color="Olive" title="b1">X</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b4">V</font> ) ) )</span><br/><b>let </b><font color="Maroon" title="c2">X</font>, <font color="Maroon" title="c3">Y</font>, <font color="Maroon" title="c4">Z</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">X</font> <a href="boolealg.html#R1" title="BOOLEALG:pred.1">=</a> <font color="Maroon" title="c3">Y</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">Z</font> iff ( <font color="Maroon" title="c3">Y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">Z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">X</font> &amp; ( for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c3">Y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">V</font> &amp; <font color="Maroon" title="c4">Z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">V</font> holds <br/><font color="Maroon" title="c2">X</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">V</font> ) ) )</span><br/>







<b>thus </b><a NAME="E1:9"/>
( <font color="Maroon" title="c2">X</font> <a href="boolealg.html#R1" title="BOOLEALG:pred.1">=</a> <font color="Maroon" title="c3">Y</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">Z</font> implies ( <font color="Maroon" title="c3">Y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">Z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">X</font> &amp; ( for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c3">Y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">V</font> &amp; <font color="Maroon" title="c4">Z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">V</font> holds <br/><font color="Maroon" title="c2">X</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">V</font> ) ) )
 <b>by </b><i><a class="ref" href="filter_0.html#T6" title="FILTER_0:th.6">FILTER_0:6</a>, <a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">Y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">X</font> &amp; <font color="Maroon" title="c4">Z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">X</font> &amp; ( for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c3">Y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">V</font> &amp; <font color="Maroon" title="c4">Z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">V</font> holds <br/><font color="Maroon" title="c2">X</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">V</font> ) implies <font color="Maroon" title="c2">X</font> <a href="boolealg.html#R1" title="BOOLEALG:pred.1">=</a> <font color="Maroon" title="c3">Y</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">Z</font> )</span><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:9"/><i><font color="Green" title="E3">A1</font></i>: 
<font color="Maroon" title="c3">Y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E3:9"/><i><font color="Green" title="E4">A2</font></i>: 
<font color="Maroon" title="c4">Z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E4:9"/><i><font color="Green" title="E5">A3</font></i>: 
for <font color="Olive" title="b1">V</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st <font color="Maroon" title="c3">Y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">V</font> &amp; <font color="Maroon" title="c4">Z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">V</font> holds <br/><font color="Maroon" title="c2">X</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">X</font> <a href="boolealg.html#R1" title="BOOLEALG:pred.1">=</a> <font color="Maroon" title="c3">Y</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">Z</font></span><br/>

<a NAME="E5:9"/>
( <font color="Maroon" title="c3">Y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">Y</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">Z</font> &amp; <font color="Maroon" title="c4">Z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">Y</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">Z</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></i>;<br/>
<a NAME="E6:9"/><b>then </b><i><font color="Green" title="E6">A4</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c3">Y</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">Z</font>
 
<b>by </b><i><a class="txt" href="boolealg.html#E4:9"><i><font color="Green" title="E5">A3</font></i></a></i>;<br/>
<a NAME="E7:9"/>
<font color="Maroon" title="c3">Y</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">Z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="boolealg.html#E2:9"><i><font color="Green" title="E3">A1</font></i></a>, <a class="txt" href="boolealg.html#E3:9"><i><font color="Green" title="E4">A2</font></i></a>, <a class="ref" href="filter_0.html#T6" title="FILTER_0:th.6">FILTER_0:6</a></i>;<br/>
<b>hence </b><a NAME="E8:9"/>
<font color="Maroon" title="c2">X</font> <a href="boolealg.html#R1" title="BOOLEALG:pred.1">=</a> <font color="Maroon" title="c3">Y</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c4">Z</font>
 <b>by </b><i><a class="txt" href="boolealg.html#E6:9"><i><font color="Green" title="E6">A4</font></i></a>, <a class="ref" href="lattices.html#T26" title="LATTICES:th.26">LATTICES:26</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>
