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

<b>set </b><font color="Maroon">G</font> =  <a href="normform.html#K12" target="_self">NormForm</a> <font color="Maroon">A</font>;<br/>
<b>thus </b><a NAME="E1:77_1_1"/>
 <a href="normform.html#K12" target="_self">NormForm</a> <font color="Maroon">A</font> is <a href="lattices.html#V11">distributive</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normform.html#K12" target="_self">NormForm</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">v</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normform.html#K12" target="_self">NormForm</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">w</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normform.html#K12" target="_self">NormForm</a> <font color="Maroon">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D11">LATTICES:def 11</a><br/>

<b>reconsider </b><font color="Maroon">K</font> = <font color="Maroon">u</font>, <font color="Maroon">L</font> = <font color="Maroon">v</font>, <font color="Maroon">M</font> = <font color="Maroon">w</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K8" target="_self">Normal_forms_on</a> <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<b>thus </b><font color="Maroon">u</font> <a href="lattices.html#K4">"/\"</a> <span class="p1">(<span class="default"><font color="Maroon">v</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">w</font></span>)</span> = 
the <a href="lattices.html#U1">L_meet</a> of <span class="p1">(<span class="default"><a href="normform.html#K12" target="_self">NormForm</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">K</font>,<span class="p1">(<span class="default">the <a href="lattices.html#U2">L_join</a> of <span class="p2">(<span class="default"><a href="normform.html#K12" target="_self">NormForm</a> <font color="Maroon">A</font></span>)</span> <a href="binop_1.html#K1">.</a> <font color="Maroon">L</font>,<font color="Maroon">M</font></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">v</font></span>)</span> <a href="lattices.html#K3">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">w</font></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E2:77_1_1"/>
 <a href="normform.html#K12" target="_self">NormForm</a> <font color="Maroon">A</font> is <a href="lattices.html#V13">lower-bounded</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1_1_2"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon">E</font> =  <a href="xboole_0.html#K1">{}</a>  as    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K8" target="_self">Normal_forms_on</a> <font color="Maroon">A</font> <b>by </b><i><a class="ref" href="normform.html#T22" target="_self">Th22</a></i>;<br/>
<b>reconsider </b><font color="Maroon">e</font> = <font color="Maroon">E</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normform.html#K12" target="_self">NormForm</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="lattices.html#D13">LATTICES:def 13</a><br/>

<b>let </b><font color="Maroon">u</font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="normform.html#K12" target="_self">NormForm</a> <font color="Maroon">A</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">K</font> = <font color="Maroon">u</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="normform.html#K8" target="_self">Normal_forms_on</a> <font color="Maroon">A</font> <b>by </b><i/>;<br/>
<font color="Maroon">e</font> <a href="lattices.html#K3">"\/"</a> <font color="Maroon">u</font> = 
 <a href="normform.html#K9" target="_self">mi</a> <span class="p1">(<span class="default"><font color="Maroon">E</font> <a href="setwiseo.html#K5">\/</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">u</font>
<b>by </b><i/>
;<br/>
<a NAME="E5:77_1_1_2"/><b>then </b>
( <font color="Maroon">e</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">u</font> <a href="lattices.html#K4">"/\"</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 
<b>by </b><i><a class="ref" href="lattices.html#D9">LATTICES:def 9</a></i>;<br/>
<b>hence </b><a NAME="E6:77_1_1_2"/>
( <font color="Maroon">e</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> &amp; <font color="Maroon">u</font> <a href="lattices.html#K2">"/\"</a> <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <font color="Maroon">e</font> )
 ;<br/>


</div><b>end;</b></div>


</div>
