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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#L3" title="LATTICES:struct.3">LattStr</a> ;<br/>

<b>assume </b><a NAME="E1:7"/><i><font color="Green" title="E8">A1</font></i>: 
( <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V2" title="ROBBINS3:attr.2">meet-Associative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V1" title="ROBBINS3:attr.1">join-Associative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="robbins3.html#V3" title="ROBBINS3:attr.3">meet-Absorbing</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V9" title="LATTICES:attr.9">join-absorbing</a> )
 ;<br/>

<a NAME="E2:7"/><b>then </b><i><font color="Green" title="E9">A2</font></i>: 
( <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V4" title="LATTICES:attr.4">join-commutative</a> )
 
<b>by </b><i><a class="ref" href="robbins3.html#T2" target="_self" title="ROBBINS3:th.2">Th2</a></i>;<br/>
<a NAME="E3:7"/><i><font color="Green" title="E10">A3</font></i>: 
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  <font color="Olive" title="b1">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b3">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font></span>)</span> = 
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>
<b>by </b><i><a class="txt" href="robbins3.html#E2:7"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="lattices.html#D6" title="LATTICES:def.6">LATTICES:def 6</a></i>
<br/>.= 
<font color="Maroon" title="c4">c</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span>
<b>by </b><i><a class="txt" href="robbins3.html#E1:7"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="robbins3.html#D2" target="_self" title="ROBBINS3:def.2">Def2</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="txt" href="robbins3.html#E2:7"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="lattices.html#D6" title="LATTICES:def.6">LATTICES:def 6</a></i>
;<br/>
<b>hence </b><a NAME="E2:7_1"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">c</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:7"/>
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  <font color="Olive" title="b1">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b3">z</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b3">z</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>;<br/>

<font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">c</font></span>)</span> = 
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">c</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span>
<b>by </b><i><a class="txt" href="robbins3.html#E2:7"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="lattices.html#D4" title="LATTICES:def.4">LATTICES:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c4">c</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span>
<b>by </b><i><a class="txt" href="robbins3.html#E1:7"><i><font color="Green" title="E8">A1</font></i></a>, <a class="ref" href="robbins3.html#D1" target="_self" title="ROBBINS3:def.1">Def1</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">c</font>
<b>by </b><i><a class="txt" href="robbins3.html#E2:7"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="lattices.html#D4" title="LATTICES:def.4">LATTICES:def 4</a></i>
;<br/>
<b>hence </b><a NAME="E2:7_2"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font></span>)</span> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c4">c</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:7"/>
( <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V7" title="LATTICES:attr.7">meet-associative</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="lattices.html#V5" title="LATTICES:attr.5">join-associative</a> )
 <b>by </b><i><a class="txt" href="robbins3.html#E3:7"><i><font color="Green" title="E10">A3</font></i></a>, <a class="ref" href="lattices.html#D5" title="LATTICES:def.5">LATTICES:def 5</a>, <a class="ref" href="lattices.html#D7" title="LATTICES:def.7">LATTICES:def 7</a></i>;<br/>


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