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

<b>let </b><font color="Maroon">N</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_2.html#V2">meet-continuous</a> <a href="waybel19.html#V2">Lawson</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">S</font> be  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</font>;<br/>





<a NAME="E1:56"/><i><font color="Green">E32</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">N</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</font> #) <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> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">l</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">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="waybel19.html#K2">lambda</a> <font color="Maroon">N</font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">s</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">J</font></span>)</span> where <font color="Olive">J</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">J</font> &amp; <font color="Olive">J</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font> ) </span>}</span> ;<br/>
<b>thus </b><a NAME="E2:56"/>
<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">J</font></span>)</span> where <font color="Olive">J</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">J</font> &amp; <font color="Olive">J</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font> ) </span>}</span>  <a href="tarski.html#R1">c=</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">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="waybel19.html#K2">lambda</a> <font color="Maroon">N</font> ) </span>}</span> 
 <b>by </b><i/>; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><br/>

<b>let </b><font color="Maroon">k</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="E3:56"/>
<font color="Maroon">k</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">A</font></span>)</span> where <font color="Olive">A</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">A</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R2">in</a>  <a href="waybel19.html#K2">lambda</a> <font color="Maroon">N</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">A</font> being   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font><b> such that </b><br/><a NAME="E5:56"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">A</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R2">in</a>  <a href="waybel19.html#K2">lambda</a> <font color="Maroon">N</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">J</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i><a class="txt" href="waybel30.html#E1:56"><i><font color="Green">E32</font></i></a></i>;<br/>
<a NAME="E7:56"/><i><font color="Green">E35</font></i>: 
 <a href="yellow_0.html#R2">ex_inf_of</a> <font color="Maroon">J</font>,<font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<b>then </b><i><font color="Green">E36</font></i>:  <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">A</font> = 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <font color="Maroon">J</font>
<b>by </b><i><a class="txt" href="waybel30.html#E1:56"><i><font color="Green">E32</font></i></a>, <a class="ref" href="yellow_0.html#T27">YELLOW_0:27</a></i>
<br/>.= 
 <a href="yellow_0.html#NK4" target="_self">inf</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">J</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel30.html#E5:56"><i><font color="Green">E34</font></i></a>, <a class="ref" href="waybel_0.html#T38">WAYBEL_0:38</a></i>
;<br/>
<a NAME="E9:56"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">J</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">J</font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#T16">WAYBEL_0:16</a></i>;<br/>
<a NAME="E10:56"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:56"/><b>then </b>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">J</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<a NAME="E12:56"/><b>then </b>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">J</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="waybel14.html#T24">WAYBEL14:24</a></i>;<br/>
<b>hence </b><a NAME="E13:56"/>
<font color="Maroon">k</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">J</font></span>)</span> where <font color="Olive">J</font> is   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> : ( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Olive">J</font> &amp; <font color="Olive">J</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font> ) </span>}</span> 
 <b>by </b><i>, <a class="txt" href="waybel30.html#E7:56"><i><font color="Green">E35</font></i></a>, <a class="txt" href="waybel30.html#E8:56"><i><font color="Green">E36</font></i></a></i>;<br/>


</div>
