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<b>let </b><font color="Maroon">L</font> be  <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#NM6">Semilattice</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:77_1"/>
<font color="Maroon">L</font> is <a href="waybel_2.html#V2" target="_self">meet-continuous</a>
 ;<br/><a NAME="E2:77_1"/><b>then </b>
<font color="Maroon">L</font> is <a href="waybel_2.html#V1" target="_self">satisfying_MC</a>
 <b>by </b><i/>;<br/><a NAME="E3:77_1"/><b>then </b>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><br/> for <font color="Olive">D</font> being non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">D</font> holds <br/><font color="Olive">x</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">x</font></span><a href="domain_1.html#K6">}</a></span> <a href="yellow_4.html#K4">"/\"</a> <font color="Olive">D</font></span>)</span>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:77_1"/>
 <a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">L</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:77"/>
 <a href="waybel_2.html#K4" target="_self">inf_op</a> <font color="Maroon">L</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 ;<br/>

<a NAME="E3:77"/><b>then </b>
for <font color="Olive">D1</font>, <font color="Olive">D2</font> being non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> holds  <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">D1</font></span>)</span> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Olive">D2</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Olive">D1</font> <a href="yellow_4.html#K4">"/\"</a> <font color="Olive">D2</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:77"/>
( <font color="Maroon">L</font> is <a href="waybel_0.html#V24">up-complete</a> &amp; <font color="Maroon">L</font> is <a href="waybel_2.html#V1" target="_self">satisfying_MC</a> )
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_2.html#D7" target="_self">WAYBEL_2:def 7</a><br/>


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