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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="lattices.html#V4">join-commutative</a> <a href="lattices.html#V5">join-associative</a> <a href="robbins1.html#V5" target="_self">Huntington</a> <a href="robbins1.html#V6" target="_self">join-idempotent</a>  <a href="robbins1.html#L2" target="_self">ComplLattStr</a> ;<br/><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>



<a NAME="E1:31"/>
for <font color="Olive">b</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="Maroon">a</font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K6" target="_self">*'</a> <font color="Olive">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/>

<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K6" target="_self">*'</a> <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p5">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K5" target="_self">+</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> 
<b>by </b><i><a class="ref" href="lattices.html#D5">LATTICES:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> 
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K5" target="_self">+</a> <font color="Maroon">a</font></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> 
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E2:31_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span> <a href="robbins1.html#K6" target="_self">*'</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> 
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:31"/>
 <a href="robbins1.html#K7" target="_self">Bot</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="robbins1.html#K3" target="_self">`</a> </span>)</span></span>)</span> <a href="robbins1.html#K3" target="_self">`</a> 
 <b>by </b><i/>;<br/>


</div>
