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<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <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> ) iff <font color="Maroon" title="c1">L</font> is <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a> )</span><br/>

<b>thus </b><a NAME="E1:35"/>
( <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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:35_1"/><i><font color="Green" title="E26">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="E2:35_1"/><i><font color="Green" title="E27">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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">L</font> is <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a></span><br/>

<a NAME="E3:35_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#E1:35_1"><i><font color="Green" title="E26">A1</font></i></a></i>;<br/>
<a NAME="E4:35_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="E5:35_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#E1:35_1"><i><font color="Green" title="E26">A1</font></i></a>, <a class="txt" href="waybel15.html#E2:35_1"><i><font color="Green" title="E27">A2</font></i></a>, <a class="ref" href="waybel_6.html#T39" title="WAYBEL_6:th.39">WAYBEL_6:39</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>thus </b><a NAME="E2:35"/>
( <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> ) )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:35_2"/>
<font color="Maroon" title="c1">L</font> is <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> )</span><br/>

<a NAME="E2:35_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="E3:35_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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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


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