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

<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/>

<b>thus </b><a NAME="E1:60"/>
( <font color="Maroon">L</font> is <a href="waybel_5.html#V1">completely-distributive</a> implies ( <font color="Maroon">L</font> is <a href="waybel_3.html#V3">continuous</a> &amp; ( for <font color="Olive">l</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">X</font> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> holds <br/><font color="Olive">x</font> is <a href="waybel_6.html#V6" target="_self">co-prime</a> ) ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="60_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:60_1"/>
<font color="Maroon">L</font> is <a href="waybel_5.html#V1">completely-distributive</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">L</font> = <font color="Maroon">L</font> as  <a href="waybel_5.html#V1">completely-distributive</a> <a href="waybel_0.html#NM8">LATTICE</a> ;<br/>
<a NAME="E3:60_1"/>
( <font color="Maroon">L</font> is <a href="waybel_1.html#V2">distributive</a> &amp; <font color="Maroon">L</font> is <a href="waybel_3.html#V3">continuous</a> &amp; <font color="Maroon">L</font> <a href="lattice3.html#K7">~</a>  is <a href="waybel_3.html#V3">continuous</a> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:60_1"/>
( <font color="Maroon">L</font> is <a href="waybel_3.html#V3">continuous</a> &amp; ( for <font color="Olive">l</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">X</font> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> holds <br/><font color="Olive">x</font> is <a href="waybel_6.html#V6" target="_self">co-prime</a> ) ) ) )
 <b>by </b><i/>;<br/>


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

<b>thus </b><a NAME="E2:60"/>
( <font color="Maroon">L</font> is <a href="waybel_3.html#V3">continuous</a> &amp; ( for <font color="Olive">l</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  ex <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> st <br/>( <font color="Olive">l</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">X</font> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">X</font> holds <br/><font color="Olive">x</font> is <a href="waybel_6.html#V6" target="_self">co-prime</a> ) ) ) implies <font color="Maroon">L</font> is <a href="waybel_5.html#V1">completely-distributive</a> )
 <b>by </b><i/>;<br/>


</div>
