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<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM6">Semilattice</a>;<br/>

<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:80_1"/>
<font color="Maroon">L</font> is <a href="waybel_2.html#V2" target="_self">meet-continuous</a>
 ;<br/><a NAME="E2:80_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">N</font> being non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#NM1">prenet</a> of <font color="Maroon">L</font>  st <font color="Olive">N</font> is <a href="waybel_0.html#V10">eventually-directed</a> holds <br/><font color="Olive">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K1" target="_self">sup</a> <font color="Olive">N</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"><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> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <font color="Olive">N</font>,<font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_2.html#T6" target="_self">Th6</a></i>;<br/><b>hence </b><a NAME="E3:80_1"/>
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">J</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Olive">J</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> holds  <font color="Olive">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_2.html#NK6" target="_self">Sup</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel_2.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="waybel_2.html#K3" target="_self">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_2.html#K2" target="_self">FinSups</a> <font color="Olive">f</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="waybel_2.html#T4" target="_self">Th4</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:80"/>
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">J</font> being   <a href="hidden.html#M1">set</a> <br/> for <font color="Olive">f</font> being  <a href="funct_2.html#NM1">Function</a> of <font color="Olive">J</font>,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> holds  <font color="Olive">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="yellow_2.html#NK6" target="_self">Sup</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel_2.html#K1" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="waybel_2.html#K3" target="_self">"/\"</a> <span class="p2">(<span class="default"><a href="waybel_2.html#K2" target="_self">FinSups</a> <font color="Olive">f</font></span>)</span></span>)</span>
 ;<br/>

<a NAME="E3:80"/><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">N</font> being  <a href="waybel_0.html#NM1">prenet</a> of <font color="Maroon">L</font>  st <font color="Olive">N</font> is <a href="waybel_0.html#V10">eventually-directed</a> holds <br/><font color="Olive">x</font> <a href="lattice3.html#K12">"/\"</a> <span class="p1">(<span class="default"><a href="waybel_2.html#K1" target="_self">sup</a> <font color="Olive">N</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"><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> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="waybel_0.html#K1">netmap</a> <font color="Olive">N</font>,<font color="Maroon">L</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel_2.html#T5" target="_self">Th5</a></i>;<br/>
<b>hence </b><a NAME="E4:80"/>
( <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><a class="ref" href="waybel_2.html#T3" target="_self">Th3</a></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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