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<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>







<span class="kw">thus </span><a NAME="E1:71"/>
(  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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="71_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:71_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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:71_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="boolealg.html#D4" target="_self" title="BOOLEALG:def.4">Def4</a></span>;<br/>
<a NAME="E3:71_1"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a></span>;<br/>
<a NAME="E4:71_1"/><span class="kw">then </span>
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T39" title="LATTICES:th.39">LATTICES:39</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:71_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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="boolealg.html#D4" target="_self" title="BOOLEALG:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E2:71"/><span class="lab"><font color="Green" title="E19">A1</font></span>: 
( <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="71_2"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:71_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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71"><span class="lab"><font color="Green" title="E19">A1</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_2_1"><span class="kw">suppose </span></a><a NAME="E1:71_2_1"/><span class="lab"><font color="Green" title="E20">A2</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:71_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></span>;<br/><a NAME="E3:71_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A3</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a></span>;<br/><a NAME="E4:71_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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_2_1"><span class="lab"><font color="Green" title="E20">A2</font></span></a>, <a class="ref" href="boolealg.html#D4" target="_self" title="BOOLEALG:def.4">Def4</a></span>;<br/><a NAME="E5:71_2_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:71_2_1"><span class="lab"><font color="Green" title="E21">A3</font></span></a>, <a class="ref" href="boolealg.html#T25" target="_self" title="BOOLEALG:th.25">Th25</a></span>;<br/><span class="kw">hence </span><a NAME="E6:71_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="boolealg.html#D4" target="_self" title="BOOLEALG:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="71_2_2"><span class="kw">suppose </span></a><a NAME="E1:71_2_2"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:71_2_2"/><span class="lab"><font color="Green" title="E21">A5</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="lattices.html#D11" title="LATTICES:def.11">LATTICES:def 11</a></span>;<br/><a NAME="E3:71_2_2"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:71_2_2"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="boolealg.html#D4" target="_self" title="BOOLEALG:def.4">Def4</a></span>;<br/><a NAME="E4:71_2_2"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:71_2_2"><span class="lab"><font color="Green" title="E21">A5</font></span></a>, <a class="ref" href="boolealg.html#T27" target="_self" title="BOOLEALG:th.27">Th27</a></span>;<br/><span class="kw">hence </span><a NAME="E5:71_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="boolealg.html#D4" target="_self" title="BOOLEALG:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div>
