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<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>thus </b><a NAME="E1:66"/>
( <font color="Maroon">L</font> is <a href="waybel_5.html#V1">completely-distributive</a> implies <font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a>  is <a href="waybel_5.html#V1">completely-distributive</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="66_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_1"/>
<font color="Maroon">L</font> is <a href="waybel_5.html#V1">completely-distributive</a>
 ;<br/>

<a NAME="E2:66_1"/><b>then </b><i><font color="Green">E32</font></i>: 
<font color="Maroon">L</font> is  non <a href="struct_0.html#V3">empty</a> <a href="waybel_5.html#V1">completely-distributive</a> <a href="orders_2.html#NM1">Poset</a>
 
;<br/>
<b>hence </b><a NAME="E3:66_1"/>
<font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a>  is <a href="lattice3.html#V3">complete</a>
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_5.html#D3">WAYBEL_5:def 3</a><br/>

<b>let </b><font color="Maroon">J</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">K</font> be  <a href="relat_1.html#V2">V3</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">K</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>;<br/>





<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">F</font> as   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">K</font>,<font color="Maroon">L</font> ;<br/>
<b>thus </b> <a href="yellow_2.html#NK7" target="_self">Inf</a>  = 
 <a href="yellow_2.html#K4">\\/</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span>,<font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="yellow_7.html#T7" target="_self">Th7</a>, <a class="ref" href="yellow_7.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i><a class="ref" href="yellow_7.html#T7" target="_self">Th7</a>, </i>
<br/>.= 
 <a href="yellow_2.html#NK7" target="_self">Inf</a> 
<b>by </b><i><a class="ref" href="yellow_7.html#T7" target="_self">Th7</a>, <a class="ref" href="yellow_7.html#T5" target="_self">Th5</a></i>
<br/>.= 
 <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>,<font color="Maroon">L</font>
<b>by </b><i><a class="ref" href="yellow_7.html#T7" target="_self">Th7</a>, </i>
<br/>.= 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i><a class="ref" href="yellow_7.html#T7" target="_self">Th7</a>, <a class="ref" href="yellow_7.html#T6" target="_self">Th6</a></i>
;<br/>


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

<b>assume </b><a NAME="E2:66"/>
<font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a>  is <a href="waybel_5.html#V1">completely-distributive</a>
 ;<br/>

<a NAME="E3:66"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a>  is  non <a href="struct_0.html#V3">empty</a> <a href="waybel_5.html#V1">completely-distributive</a> <a href="orders_2.html#NM1">Poset</a>
 
;<br/>
<a NAME="E4:66"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">L</font> is  non <a href="struct_0.html#V3">empty</a> <a href="lattice3.html#V3">complete</a> <a href="orders_2.html#NM1">Poset</a>
 
<b>by </b><i/>;<br/>
<b>thus </b><a NAME="E5:66"/>
<font color="Maroon">L</font> is <a href="lattice3.html#V3">complete</a>
 <b>by </b><i><a class="ref" href="yellow_7.html#T7" target="_self">Th7</a>, </i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_5.html#D3">WAYBEL_5:def 3</a><br/>

<b>let </b><font color="Maroon">J</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">K</font> be  <a href="relat_1.html#V2">V3</a>  <a href="pboole.html#M1">ManySortedSet</a> of <font color="Maroon">J</font>;<br/><b>let </b><font color="Maroon">F</font> be   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">K</font>,<font color="Maroon">L</font>;<br/>





<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">F</font> as   <a href="waybel_5.html#NM2">DoubleIndexedSet</a> of <font color="Maroon">K</font>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span> ;<br/>
<b>thus </b> <a href="yellow_2.html#NK7" target="_self">Inf</a>  = 
 <a href="yellow_2.html#K4">\\/</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" target="_self">Sups</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow_7.html#T8" target="_self">Th8</a>, <a class="ref" href="yellow_7.html#T6" target="_self">Th6</a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i><a class="ref" href="yellow_7.html#T8" target="_self">Th8</a>, </i>
<br/>.= 
 <a href="yellow_2.html#NK7" target="_self">Inf</a> 
<b>by </b><i><a class="ref" href="yellow_7.html#T7" target="_self">Th7</a>, <a class="ref" href="yellow_7.html#T5" target="_self">Th5</a></i>
<br/>.= 
 <a href="yellow_2.html#K5">//\</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" target="_self">Infs</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">L</font> <a href="yellow_7.html#NK1" target="_self">opp</a> </span>)</span>
<b>by </b><i><a class="ref" href="yellow_7.html#T8" target="_self">Th8</a>, </i>
<br/>.= 
 <a href="yellow_2.html#NK6" target="_self">Sup</a> 
<b>by </b><i><a class="ref" href="yellow_7.html#T8" target="_self">Th8</a>, <a class="ref" href="yellow_7.html#T6" target="_self">Th6</a></i>
;<br/>


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