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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a> <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a> <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a> <a href="lattices.html#V8" title="LATTICES:attr.8">meet-absorbing</a> <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a>  <a href="lattices.html#L3" title="LATTICES:struct.3">LattStr</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>  st <font color="Olive" title="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b3">z</font> &amp; <font color="Olive" title="b2">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b3">z</font> &amp; ( for <font color="Olive" title="b4">z'</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="b1">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b4">z'</font> &amp; <font color="Olive" title="b2">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b4">z'</font> holds <br/><font color="Olive" title="b3">z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b4">z'</font> ) holds <br/><font color="Olive" title="b3">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Olive" title="b2">y</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="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">z</font> &amp; <font color="Maroon" title="c3">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">z</font> &amp; ( for <font color="Olive" title="b1">z'</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="c2">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">z'</font> &amp; <font color="Maroon" title="c3">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">z'</font> holds <br/><font color="Maroon" title="c4">z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">z'</font> ) implies <font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">y</font> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:97"/><i><font color="Green" title="E65">A1</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">z</font>
 <b>and </b><br/><a NAME="E2:97"/><i><font color="Green" title="E66">A2</font></i>: 
<font color="Maroon" title="c3">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">z</font>
 <b>and </b><br/><a NAME="E3:97"/><i><font color="Green" title="E67">A3</font></i>: 
for <font color="Olive" title="b1">z'</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="c2">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">z'</font> &amp; <font color="Maroon" title="c3">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">z'</font> holds <br/><font color="Maroon" title="c4">z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Olive" title="b1">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="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">y</font></span><br/>

<a NAME="E4:97"/><i><font color="Green" title="E68">A4</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">y</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c4">z</font>
 
<b>by </b><i><a class="txt" href="#E1:97"><i><font color="Green" title="E65">A1</font></i></a>, <a class="txt" href="#E2:97"><i><font color="Green" title="E66">A2</font></i></a>, <a class="ref" href="filter_0.html#T6" target="_self" title="FILTER_0:th.6">Th6</a></i>;<br/>
<a NAME="E5:97"/>
( <font color="Maroon" title="c2">x</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">y</font> &amp; <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#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">y</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#T22" title="LATTICES:th.22">LATTICES:22</a></i>;<br/>
<a NAME="E6:97"/><b>then </b>
<font color="Maroon" title="c4">z</font> <a href="lattices.html#R3" title="LATTICES:pred.3">[=</a> <font color="Maroon" title="c2">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">y</font>
 
<b>by </b><i><a class="txt" href="#E3:97"><i><font color="Green" title="E67">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:97"/>
<font color="Maroon" title="c4">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">x</font> <a href="lattices.html#K3" title="LATTICES:func.3">"\/"</a> <font color="Maroon" title="c3">y</font>
 <b>by </b><i><a class="txt" href="#E4:97"><i><font color="Green" title="E68">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/>


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