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

<b>let </b><font color="Maroon" title="c1">L</font> be  <a href="waybel_1.html#V11" title="WAYBEL_1:attr.11">Boolean</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/>

<b>thus </b><a NAME="E1:37"/>
( <font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> &amp; <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> implies <font color="Maroon" title="c1">L</font> is <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:37_1"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 <b>and </b><br/><a NAME="E2:37_1"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 ;<br/>

<a NAME="E3:37_1"/>
<font color="Maroon" title="c1">L</font> is  <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 
<b>by </b><i><a class="txt" href="waybel15.html#E1:37_1"><i><font color="Green" title="E30">A1</font></i></a></i>;<br/>
<a NAME="E4:37_1"/><b>then </b>
<font color="Maroon" title="c1">L</font> is  <a href="yellow_0.html#V1" title="YELLOW_0:attr.1">lower-bounded</a> <a href="waybel_0.html#V24" title="WAYBEL_0:attr.24">up-complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 
;<br/>
<b>hence </b><a NAME="E5:37_1"/>
<font color="Maroon" title="c1">L</font> is <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a>
 <b>by </b><i><a class="txt" href="waybel15.html#E1:37_1"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="waybel15.html#E2:37_1"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="waybel_6.html#T39" title="WAYBEL_6:th.39">WAYBEL_6:39</a></i>;<br/>


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

<b>thus </b><a NAME="E2:37"/>
( <font color="Maroon" title="c1">L</font> is <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a> implies ( <font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> &amp; <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="37_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:37_2"/>
<font color="Maroon" title="c1">L</font> is <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a>
 ;<br/>

<a NAME="E2:37_2"/><b>then </b>
<font color="Maroon" title="c1">L</font> is  <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>
 
;<br/>
<b>hence </b><a NAME="E3:37_2"/>
( <font color="Maroon" title="c1">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> &amp; <font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> )
 <b>by </b><i><a class="ref" href="waybel_6.html#T39" title="WAYBEL_6:th.39">WAYBEL_6:39</a></i>;<br/>


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


</div>
