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<b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">D</font> be  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 <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">S</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel25.html#D4" target="_self">WAYBEL25:def 4</a><br/>



<b>thus </b><a NAME="E1:61"/><i><font color="Green">E42</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</font>, <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">S</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<b>let </b><font color="Maroon">V</font> be  <a href="pre_topc.html#V3">open</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>;<br/>

<b>assume </b><a NAME="E2:61"/><i><font color="Green">E43</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 ;<br/>

<a NAME="E3:61"/><i><font color="Green">E44</font></i>: 
 <a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="waybel_9.html#G1">TopRelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font>,the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font> #)
 
<b>by </b><i/>;<br/>
<a NAME="E4:61"/><b>then </b><i><font color="Green">E47</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">S</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" target="_self">Omega</a> <font color="Maroon">S</font></span>)</span> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #)
 
;<br/>
<b>reconsider </b><font color="Maroon">E</font> = <font color="Maroon">D</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E6:61"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">E</font> is  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">S</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T3">WAYBEL_0:3</a></i>;<br/>
<a NAME="E7:61"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">E</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 
<b>by </b><i>, , , <a class="ref" href="yellow_0.html#T26">YELLOW_0:26</a></i>;<br/>
<b>hence </b><a NAME="E8:61"/>
<font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">V</font>
 <b>by </b><i>, <a class="ref" href="waybel11.html#D1">WAYBEL11:def 1</a></i>;<br/>


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