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<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="lattices.html#NM8" title="LATTICES:NM.8">B_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#R3" title="BOOLEALG:pred.3">meets</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="b1">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</a> <font color="Olive" title="b2">Y</font> or <font color="Olive" title="b1">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</a> <font color="Olive" title="b3">Z</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#R3" title="BOOLEALG:pred.3">meets</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="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</a> <font color="Maroon" title="c3">Y</font> or <font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</a> <font color="Maroon" title="c4">Z</font> ) )</span><br/>







<b>thus </b><a NAME="E1:51"/>
(  not <font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</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> or <font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</a> <font color="Maroon" title="c3">Y</font> or <font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</a> <font color="Maroon" title="c4">Z</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#R3" title="BOOLEALG:pred.3">meets</a> <font color="Maroon" title="c3">Y</font> or <font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</a> <font color="Maroon" title="c4">Z</font> ) implies <font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</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><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:51_1"/>
<font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</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>
 ; <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#R3" title="BOOLEALG:pred.3">meets</a> <font color="Maroon" title="c3">Y</font> or <font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</a> <font color="Maroon" title="c4">Z</font> )</span><br/>

<a NAME="E2:51_1"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><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>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="boolealg.html#D4" target="_self" title="BOOLEALG:def.4">Def4</a></i>;<br/>
<a NAME="E3:51_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">Z</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
 
<b>by </b><i><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a></i>;<br/>
<a NAME="E4:51_1"/><b>then </b>
( <font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> or <font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T39" title="LATTICES:th.39">LATTICES:39</a></i>;<br/>
<b>hence </b><a NAME="E5:51_1"/>
( <font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</a> <font color="Maroon" title="c3">Y</font> or <font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</a> <font color="Maroon" title="c4">Z</font> )
 <b>by </b><i><a class="ref" href="boolealg.html#D4" target="_self" title="BOOLEALG:def.4">Def4</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>

<b>assume </b><a NAME="E2:51"/><i><font color="Green" title="E19">A1</font></i>: 
( <font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</a> <font color="Maroon" title="c3">Y</font> or <font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</a> <font color="Maroon" title="c4">Z</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#R3" title="BOOLEALG:pred.3">meets</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/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:51_2"/>
( <font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</a> <font color="Maroon" title="c3">Y</font> or <font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</a> <font color="Maroon" title="c4">Z</font> )
 </span><b>by </b><i><a class="txt" href="boolealg.html#E2:51"><i><font color="Green" title="E19">A1</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_1"><b>suppose </b></a><a NAME="E1:51_2_1"/>
<font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</a> <font color="Maroon" title="c3">Y</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#R3" title="BOOLEALG:pred.3">meets</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/><div class="add"><a NAME="E2:51_2_1"/><b>then </b><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">Y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="boolealg.html#D4" target="_self" title="BOOLEALG:def.4">Def4</a></i>;<br/><a NAME="E3:51_2_1"/>
<font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">Y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">Z</font></span>)</span>
 <b>by </b><i><a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></i>;<br/><a NAME="E4:51_2_1"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">Y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><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>)</span>
 <b>by </b><i><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a></i>;<br/><a NAME="E5:51_2_1"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><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>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="boolealg.html#E2:51_2_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="boolealg.html#T25" target="_self" title="BOOLEALG:th.25">Th25</a></i>;<br/><b>hence </b><a NAME="E6:51_2_1"/>
<font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</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="boolealg.html#D4" target="_self" title="BOOLEALG:def.4">Def4</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="51_2_2"><b>suppose </b></a><a NAME="E1:51_2_2"/>
<font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</a> <font color="Maroon" title="c4">Z</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#R3" title="BOOLEALG:pred.3">meets</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/><div class="add"><a NAME="E2:51_2_2"/><b>then </b><i><font color="Green" title="E20">A3</font></i>: 
<font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">Z</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="boolealg.html#D4" target="_self" title="BOOLEALG:def.4">Def4</a></i>;<br/><a NAME="E3:51_2_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c4">Z</font></span>)</span> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <font color="Maroon" title="c3">Y</font></span>)</span> <a href="boolealg.html#R1" title="BOOLEALG:pred.1">=</a> <font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><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>)</span>
 <b>by </b><i><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a></i>;<br/><a NAME="E4:51_2_2"/><b>then </b>
<font color="Maroon" title="c2">X</font> <a href="lattices.html#K4" title="LATTICES:func.4">"/\"</a> <span class="p1">(<span class="default"><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>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="lattices.html#K5" title="LATTICES:func.5">Bottom</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="txt" href="boolealg.html#E2:51_2_2"><i><font color="Green" title="E20">A3</font></i></a>, <a class="ref" href="boolealg.html#T27" target="_self" title="BOOLEALG:th.27">Th27</a></i>;<br/><b>hence </b><a NAME="E5:51_2_2"/>
<font color="Maroon" title="c2">X</font> <a href="boolealg.html#R3" title="BOOLEALG:pred.3">meets</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="boolealg.html#D4" target="_self" title="BOOLEALG:def.4">Def4</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></div><b>end;</b></div>

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