<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>  <a href="lattices.html#L3" title="LATTICES:struct.3">LattStr</a> ;<br/>

<b>assume </b><a NAME="E1:5"/><i><font color="Green" title="E6">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="E2:5"/><i><font color="Green" title="E7">A2</font></i>: 
( <font color="Maroon" title="c1">L</font> is <a href="robbins1.html#V6" title="ROBBINS1:attr.6">join-idempotent</a> &amp; <font color="Maroon" title="c1">L</font> is <a href="sheffer1.html#V5" title="SHEFFER1:attr.5">meet-idempotent</a> )
 
<b>by </b><i><a class="txt" href="robbins3.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="robbins3.html#T1" target="_self" title="ROBBINS3:th.1">Th1</a></i>;<br/>
<a NAME="E3:5"/><i><font color="Green" title="E8">A3</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_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> 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> <font color="Maroon" title="c3">b</font> = 
<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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D9" title="LATTICES:def.9">LATTICES:def 9</a>, <a class="txt" href="robbins3.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">b</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="robbins3.html#D2" target="_self" title="ROBBINS3:def.2">Def2</a>, <a class="txt" href="robbins3.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">b</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="robbins3.html#E2:5"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="robbins1.html#D7" title="ROBBINS1:def.7">ROBBINS1:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="robbins3.html#D1" target="_self" title="ROBBINS3:def.1">Def1</a>, <a class="txt" href="robbins3.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="robbins3.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="lattices.html#D9" title="LATTICES:def.9">LATTICES:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E2:5_1"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c2">a</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E4:5"/>
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</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> <font color="Olive" title="b2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">y</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Olive" title="b1">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_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> 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> <font color="Maroon" title="c3">b</font> = 
<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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="robbins3.html#D3" target="_self" title="ROBBINS3:def.3">Def3</a>, <a class="txt" href="robbins3.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">b</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="robbins3.html#D1" target="_self" title="ROBBINS3:def.1">Def1</a>, <a class="txt" href="robbins3.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">b</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> <span class="p2">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="robbins3.html#E2:5"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="sheffer1.html#D9" title="SHEFFER1:def.9">SHEFFER1:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c3">b</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> <span class="p2">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">b</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c2">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="robbins3.html#D2" target="_self" title="ROBBINS3:def.2">Def2</a>, <a class="txt" href="robbins3.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c3">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="ref" href="robbins3.html#D3" target="_self" title="ROBBINS3:def.3">Def3</a>, <a class="txt" href="robbins3.html#E1:5"><i><font color="Green" title="E6">A1</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E2:5_2"/>
<font color="Maroon" title="c2">a</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">b</font> <a href="lattices.html#K1" title="LATTICES:func.1">"\/"</a> <font color="Maroon" title="c2">a</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:5"/>
( <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="txt" href="robbins3.html#E3:5"><i><font color="Green" title="E8">A3</font></i></a>, <a class="ref" href="lattices.html#D6" title="LATTICES:def.6">LATTICES:def 6</a>, <a class="ref" href="lattices.html#D4" title="LATTICES:def.4">LATTICES:def 4</a></i>;<br/>


</div>
