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

<b>let </b><font color="Maroon">L</font> be   <a href="lattices.html#NM1">Lattice</a>;<br/>

<b>assume </b><a NAME="E1:55"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Olive">b</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p2">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Olive">c</font></span>)</span></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Olive">c</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Olive">a</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive">a</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Olive">b</font></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p2">(<span class="default"><font color="Olive">b</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Olive">c</font></span>)</span></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p1">(<span class="default"><font color="Olive">c</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Olive">a</font></span>)</span>
 ;<br/>

<div><i><font color="Green">E37</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><b>assume </b><a NAME="E1:55_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c</font> <a href="lattices.html#R3" target="_self">[=</a> <font color="Maroon">a</font>
 ;<br/><b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">c</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">c</font></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">c</font></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">c</font></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="lattices.html#E1:55"><i><font color="Green">E34</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D10" target="_self">Def10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i><a class="txt" href="lattices.html#E1:55"><i><font color="Green">E34</font></i></a>, </i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">c</font>
<b>by </b><i/>
;<br/></div><b>end;</b></div>
<b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>, <font color="Maroon">c</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D11" target="_self">LATTICES:def 11</a><br/>

<a NAME="E3:55"/><i><font color="Green">E40</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#R3" target="_self">[=</a> <font color="Maroon">a</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>proof </b></a><div class="add">

<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">a</font> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">a</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i/>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D3" target="_self">LATTICES:def 3</a><br/>


</div><b>end;</b></div>
<b>thus </b><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">c</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">a</font></span>)</span></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">a</font></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">c</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">c</font></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K3" target="_self">"\/"</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="lattices.html#D10" target="_self">Def10</a></i>
<br/>.= 
<font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p3">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">a</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="txt" href="lattices.html#E1:55"><i><font color="Green">E34</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">a</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">c</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">c</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">a</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">b</font></span>)</span> <a href="lattices.html#K3" target="_self">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="lattices.html#K4" target="_self">"/\"</a> <font color="Maroon">c</font></span>)</span>
<b>by </b><i/>
;<br/>


</div>
