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

<b>let </b><font color="Maroon">L</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>assume </b><a NAME="E1:44"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">V</font></span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font> ) </span>}</span> ,<font color="Maroon">L</font>
 ;<br/>

<b>thus </b><a NAME="E2:44"/>
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> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D6">WAYBEL_3:def 6</a><br/>

<b>thus </b><a NAME="E3:44"/>
<font color="Maroon">L</font> is <a href="waybel_0.html#V24">up-complete</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D5">WAYBEL_3:def 5</a><br/>

<b>set </b><font color="Maroon">VV</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">V</font></span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font> ) </span>}</span> ;<br/>
<a NAME="E4:44"/><i><font color="Green">E37</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">V</font></span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">d</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:44_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">V</font></span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">V</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E3:44_1"/><i><font color="Green">E38</font></i>: 
(  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">V</font> <a href="hidden.html#R1">=</a> <font color="Maroon">d</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font> )
 ;<br/>
<a NAME="E4:44_1"/>
<font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:44_1"/><b>then </b>
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">V</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:44_1"/>
<font color="Maroon">d</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K1">waybelow</a> <font color="Maroon">x</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E5:44"/>
(  <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">V</font></span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font> ) </span>}</span> ,<font color="Maroon">L</font> &amp;  <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="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E6:44"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">V</font></span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font> ) </span>}</span> ,<font color="Maroon">L</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>, <a class="ref" href="yellow_0.html#T34">YELLOW_0:34</a></i>;<br/>
<a NAME="E7:44"/>
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Olive">V</font></span>)</span> where <font color="Olive">V</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">L</font> ) </span>}</span> ,<font color="Maroon">L</font> &amp;  <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="waybel14.html#T1" target="_self">Th1</a>, </i>;<br/>
<b>hence </b><a NAME="E8:44"/>
<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="orders_2.html#T25">ORDERS_2:25</a></i>;<br/>


</div>
