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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a>  <a href="lattices.html#L3">LattStr</a> ;<br/>

<b>assume </b><a NAME="E1:6"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="robbins3.html#V2" target="_self">meet-Associative</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="robbins3.html#V1" target="_self">join-Associative</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="robbins3.html#V3" target="_self">meet-Absorbing</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V9">join-absorbing</a> )
 ;<br/>

<a NAME="E2:6"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V6">meet-commutative</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V4">join-commutative</a> )
 
<b>by </b><i><a class="txt" href="robbins3.html#E1:6"><i><font color="Green">E7</font></i></a>, <a class="ref" href="robbins3.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E3:6"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> holds <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<font color="Maroon">c<sub>3</sub></font> = 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="robbins3.html#D3" target="_self">Def3</a>, <a class="txt" href="robbins3.html#E1:6"><i><font color="Green">E7</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>3</sub></font> <a href="lattices.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="robbins3.html#E2:6"><i><font color="Green">E8</font></i></a>, <a class="ref" href="lattices.html#D6">LATTICES:def 6</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="ref" href="lattices.html#D4">LATTICES:def 4</a>, <a class="txt" href="robbins3.html#E2:6"><i><font color="Green">E8</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:6_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="lattices.html#K1">"\/"</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:6"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V8">meet-absorbing</a>
 <b>by </b><i><a class="ref" href="lattices.html#D8">LATTICES:def 8</a></i>;<br/>


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