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

<b>let </b><font color="Maroon" title="c1">K</font>, <font color="Maroon" title="c2">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="E1:40"/><i><font color="Green" title="E16">A1</font></i>: 
(  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">K</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c1">K</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c1">K</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="lattices.html#G3" title="LATTICES:aggr.3">LattStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>,the <a href="lattices.html#U2" title="LATTICES:sel.2">L_join</a> of <font color="Maroon" title="c2">L</font>,the <a href="lattices.html#U1" title="LATTICES:sel.1">L_meet</a> of <font color="Maroon" title="c2">L</font> #) &amp; <font color="Maroon" title="c1">K</font> is <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a> )
 ;<br/>

<a NAME="E2:40"/>
<font color="Maroon" title="c2">L</font> is <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D6" title="LATTICES:def.6">LATTICES:def 6</a><br/>

<b>reconsider </b><font color="Maroon" title="c5">x'</font> = <font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c6">y'</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">K</font> <b>by </b><i><a class="txt" href="robbins3.html#E1:40"><i><font color="Green" title="E16">A1</font></i></a></i>;<br/>
<a NAME="E2:40_1"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon" title="c5">x'</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c6">y'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y'</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c5">x'</font>
 
<b>by </b><i><a class="txt" href="robbins3.html#E1:40"><i><font color="Green" title="E16">A1</font></i></a>, <a class="ref" href="lattices.html#D6" title="LATTICES:def.6">LATTICES:def 6</a></i>;<br/>
<a NAME="E3:40_1"/>
<font color="Maroon" title="c3">x</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="lattices.html#K2" title="LATTICES:func.2">"/\"</a> <font color="Maroon" title="c3">x</font>
 
<b>by </b><i><a class="txt" href="robbins3.html#E1:40"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="robbins3.html#E2:40_1"><i><font color="Green" title="E17">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E4:40_1"/>
<font color="Maroon" title="c3">x</font> <a href="robbins3.html#NK2" title="ROBBINS3:NK.2">|^|</a> <font color="Maroon" title="c4">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">y</font> <a href="robbins3.html#NK2" title="ROBBINS3:NK.2">|^|</a> <font color="Maroon" title="c3">x</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:40"/>
<font color="Maroon" title="c2">L</font> is <a href="lattices.html#V6" title="LATTICES:attr.6">meet-commutative</a>
 ;<br/>


</div>
