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

<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/>

<b>thus </b><a NAME="E1:66"/>
( <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> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  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="66_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:66_1"/>
<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="E2:66_1"/><b>then </b><i><font color="Green" title="E37">A1</font></i>: 
<font color="Maroon" title="c1">L</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>
 
;<br/>
<b>hence </b><a NAME="E3:66_1"/>
<font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>
 <b>by </b><i><a class="ref" href="yellow_7.html#T17" target="_self" title="YELLOW_7:th.17">Th17</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_5.html#D3" title="WAYBEL_5:def.3">WAYBEL_5:def 3</a><br/>

<b>let </b><font color="Maroon" title="c2">J</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">K</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V3</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c2">J</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c3">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>;<br/>





<b>reconsider </b><font color="Maroon" title="c5">G</font> = <font color="Maroon" title="c4">F</font> as   <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">L</font> ;<br/>
<b>thus </b> <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  = 
 <a href="yellow_2.html#K4" title="YELLOW_2:func.4">\\/</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span>,<font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="yellow_7.html#E2:66_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="yellow_7.html#T49" target="_self" title="YELLOW_7:th.49">Th49</a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
<b>by </b><i><a class="txt" href="yellow_7.html#E2:66_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="yellow_7.html#T50" target="_self" title="YELLOW_7:th.50">Th50</a></i>
<br/>.= 
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
<b>by </b><i><a class="txt" href="yellow_7.html#E2:66_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="yellow_7.html#T48" target="_self" title="YELLOW_7:th.48">Th48</a></i>
<br/>.= 
 <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span>,<font color="Maroon" title="c1">L</font>
<b>by </b><i><a class="txt" href="yellow_7.html#E2:66_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="yellow_7.html#T50" target="_self" title="YELLOW_7:th.50">Th50</a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
<b>by </b><i><a class="txt" href="yellow_7.html#E2:66_1"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="yellow_7.html#T49" target="_self" title="YELLOW_7:th.49">Th49</a></i>
;<br/>


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

<b>assume </b><a NAME="E2:66"/>
<font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a>
 ;<br/>

<a NAME="E3:66"/><b>then </b><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a>  is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="waybel_5.html#V1" title="WAYBEL_5:attr.1">completely-distributive</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>
 
;<br/>
<a NAME="E4:66"/><b>then </b><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c1">L</font> is  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>
 
<b>by </b><i><a class="ref" href="yellow_7.html#T17" target="_self" title="YELLOW_7:th.17">Th17</a></i>;<br/>
<b>thus </b><a NAME="E5:66"/>
<font color="Maroon" title="c1">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>
 <b>by </b><i><a class="txt" href="yellow_7.html#E3:66"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="yellow_7.html#T17" target="_self" title="YELLOW_7:th.17">Th17</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_5.html#D3" title="WAYBEL_5:def.3">WAYBEL_5:def 3</a><br/>

<b>let </b><font color="Maroon" title="c2">J</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c3">K</font> be  <a href="relat_1.html#V2" title="RELAT_1:attr.2">V3</a>  <a href="pboole.html#M1" title="PBOOLE:mode.1">ManySortedSet</a> of <font color="Maroon" title="c2">J</font>;<br/><b>let </b><font color="Maroon" title="c4">F</font> be   <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c3">K</font>,<font color="Maroon" title="c1">L</font>;<br/>





<b>reconsider </b><font color="Maroon" title="c5">G</font> = <font color="Maroon" title="c4">F</font> as   <a href="waybel_5.html#NM2" title="WAYBEL_5:NM.2">DoubleIndexedSet</a> of <font color="Maroon" title="c3">K</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span> ;<br/>
<b>thus </b> <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a>  = 
 <a href="yellow_2.html#K4" title="YELLOW_2:func.4">\\/</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK6" title="WAYBEL_5:NK.6">Sups</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>
<b>by </b><i><a class="txt" href="yellow_7.html#E4:66"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="yellow_7.html#T49" target="_self" title="YELLOW_7:th.49">Th49</a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
<b>by </b><i><a class="txt" href="yellow_7.html#E4:66"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="yellow_7.html#T50" target="_self" title="YELLOW_7:th.50">Th50</a></i>
<br/>.= 
 <a href="yellow_2.html#NK7" title="YELLOW_2:NK.7">Inf</a> 
<b>by </b><i><a class="txt" href="yellow_7.html#E3:66"><i><font color="Green" title="E37">A2</font></i></a>, <a class="ref" href="yellow_7.html#T48" target="_self" title="YELLOW_7:th.48">Th48</a></i>
<br/>.= 
 <a href="yellow_2.html#K5" title="YELLOW_2:func.5">//\</a> <span class="p1">(<span class="default"><a href="waybel_5.html#NK9" title="WAYBEL_5:NK.9">Infs</a> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">L</font> <a href="yellow_7.html#NK1" title="YELLOW_7:NK.1">opp</a> </span>)</span>
<b>by </b><i><a class="txt" href="yellow_7.html#E4:66"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="yellow_7.html#T50" target="_self" title="YELLOW_7:th.50">Th50</a></i>
<br/>.= 
 <a href="yellow_2.html#NK6" title="YELLOW_2:NK.6">Sup</a> 
<b>by </b><i><a class="txt" href="yellow_7.html#E4:66"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="yellow_7.html#T49" target="_self" title="YELLOW_7:th.49">Th49</a></i>
;<br/>


</div>
