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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/>

<b>assume </b><a NAME="E1:2"/><i><font color="Green">E34</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> (  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is   <a href="waybel_0.html#NM3">Ideal</a> of <font color="Maroon">L</font> &amp; <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"><a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font></span>)</span> &amp; ( for <font color="Olive">I</font> being  <a href="waybel_0.html#NM3">Ideal</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">I</font> holds <br/> <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> <a href="tarski.html#R1">c=</a> <font color="Olive">I</font> ) )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>;<br/><a NAME="E1:2_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">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"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E2:2_1"/>
( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font> is <a href="waybel_0.html#V1">directed</a> )
 <b>by </b><i/>;<br/><a NAME="E3:2_1"/><b>then </b><i><font color="Green">E38</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font>,<font color="Maroon">L</font>
 <b>by </b><i><a class="ref" href="waybel_0.html#T75">WAYBEL_0:75</a></i>;<br/><a NAME="E4:2_1"/>
 <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font> <a href="lattice3.html#R2">is_&lt;=_than</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="waybel_3.html#T9">WAYBEL_3:9</a></i>;<br/><a NAME="E5:2_1"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/><b>hence </b><a NAME="E6:2_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E3:2"/><b>then </b><i><font color="Green">E39</font></i>: 
<font color="Maroon">L</font> is <a href="waybel_3.html#V2">satisfying_axiom_of_approximation</a>
 
<b>by </b><i><a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a></i>;<br/>
<a NAME="E4:2"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds <br/> ( not  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="waybel_3.html#K1">waybelow</a> <font color="Olive">x</font> is <a href="waybel_0.html#V1">directed</a> )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:2"/>
<font color="Maroon">L</font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i>, <a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a></i>;<br/>


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