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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</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="E1:39"/><i><font color="Green">E15</font></i>: 
(  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>1</sub></font> #) <a href="hidden.html#R1">=</a>  <a href="lattices.html#G3">LattStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>2</sub></font>,the <a href="lattices.html#U1">L_meet</a> of <font color="Maroon">c<sub>2</sub></font> #) &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="lattices.html#V4">join-commutative</a> )
 ;<br/>

<a NAME="E2:39"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="lattices.html#V4">join-commutative</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>2</sub></font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D4">LATTICES:def 4</a><br/>

<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>4</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="robbins3.html#E1:39"><i><font color="Green">E15</font></i></a></i>;<br/>
<a NAME="E2:39_1"/><i><font color="Green">E16</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="robbins3.html#NK1" target="_self">|_|</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="robbins3.html#NK1" target="_self">|_|</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="robbins3.html#E1:39"><i><font color="Green">E15</font></i></a>, <a class="ref" href="lattices.html#D4">LATTICES:def 4</a></i>;<br/>
<font color="Maroon">c<sub>3</sub></font> <a href="robbins3.html#NK1" target="_self">|_|</a> <font color="Maroon">c<sub>4</sub></font> = 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>2</sub></font> <a href="binop_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>

<br/>.= 
<font color="Maroon">c<sub>5</sub></font> <a href="robbins3.html#NK1" target="_self">|_|</a> <font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="txt" href="robbins3.html#E1:39"><i><font color="Green">E15</font></i></a></i>
<br/>.= 
the <a href="lattices.html#U2">L_join</a> of <font color="Maroon">c<sub>1</sub></font> <a href="binop_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="robbins3.html#E2:39_1"><i><font color="Green">E16</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="robbins3.html#NK1" target="_self">|_|</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="robbins3.html#E1:39"><i><font color="Green">E15</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E4:39_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="robbins3.html#NK1" target="_self">|_|</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="robbins3.html#NK1" target="_self">|_|</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:39"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="lattices.html#V4">join-commutative</a>
 ;<br/>


</div>
