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<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>consider </b><font color="Maroon">S</font> being  <a href="lattice3.html#V3">complete</a> <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">N</font>;<br/>
<a NAME="E2:63"/><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">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</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">N</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">N</font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:63_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">N</font> is <a href="waybel_3.html#V3">continuous</a>
 ;<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="E2:63_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">S</font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i>, <a class="txt" href="waybel30.html#E1:63_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="yellow12.html#T15">YELLOW12:15</a></i>;<br/><a NAME="E3:63_1"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="yellow_1.html#K2">InclPoset</a> <span class="p1">(<span class="default"><a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font></span>)</span> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i><a class="ref" href="waybel14.html#T36">WAYBEL14:36</a></i>;<br/><a NAME="E4:63_1"/><i><font color="Green">E37</font></i>: 
 <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">X</font></span>)</span> where <font color="Olive">X</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">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font> ) </span>}</span> ,<font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><a NAME="E5:63_1"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>  ex <font color="Olive">J</font> being   <a href="yellow_8.html#M1">Basis</a> of <font color="Olive">x</font> st <br/>for <font color="Olive">Y</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font>  st <font color="Olive">Y</font> <a href="hidden.html#R2">in</a> <font color="Olive">J</font> holds <br/>( <font color="Olive">Y</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Olive">Y</font> is <a href="waybel_0.html#V2">filtered</a> )
 <b>by </b><i><a class="txt" href="waybel30.html#E2:63_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="waybel14.html#T35">WAYBEL14:35</a></i>;<br/><b>hence </b><font color="Maroon">x</font> = 
 <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">X</font></span>)</span> where <font color="Olive">X</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">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font> ) </span>}</span> ,<font color="Maroon">S</font>
<b>by </b><i>, <a class="txt" href="waybel30.html#E3:63_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="waybel14.html#T37">WAYBEL14:37</a></i>
<br/>.= 
 <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">X</font></span>)</span> where <font color="Olive">X</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">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font> ) </span>}</span> ,<font color="Maroon">N</font>
<b>by </b><i>, <a class="txt" href="waybel30.html#E4:63_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="yellow_0.html#T26">YELLOW_0:26</a></i>
<br/>.= 
 <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">N</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="waybel19.html#K2">lambda</a> <font color="Maroon">N</font> ) </span>}</span> ,<font color="Maroon">N</font>
<b>by </b><i/>
;<br/><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E4:63"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">N</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">N</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="waybel19.html#K2">lambda</a> <font color="Maroon">N</font> ) </span>}</span> ,<font color="Maroon">N</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="63_2"><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">S</font>;<br/><a NAME="E1:63_2"/><i><font color="Green">E45</font></i>: 
 <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">X</font></span>)</span> where <font color="Olive">X</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">X</font> &amp; <font color="Olive">X</font> <a href="hidden.html#R2">in</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">S</font> ) </span>}</span> ,<font color="Maroon">N</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/><b>thus </b><font color="Maroon">x</font> = 
 <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">N</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="waybel19.html#K2">lambda</a> <font color="Maroon">N</font> ) </span>}</span> ,<font color="Maroon">N</font>
<b>by </b><i>, </i>
<br/>.= 
 <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">S</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">S</font> ) </span>}</span> ,<font color="Maroon">N</font>
<b>by </b><i>, </i>
<br/>.= 
 <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">S</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">S</font> ) </span>}</span> ,<font color="Maroon">S</font>
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T26">YELLOW_0:26</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E6:63"/><b>then </b>
<font color="Maroon">S</font> is <a href="waybel_3.html#V3">continuous</a>
 
<b>by </b><i><a class="ref" href="waybel14.html#T38">WAYBEL14:38</a></i>;<br/>
<b>hence </b><a NAME="E7:63"/>
<font color="Maroon">N</font> is <a href="waybel_3.html#V3">continuous</a>
 <b>by </b><i>, <a class="ref" href="yellow12.html#T15">YELLOW12:15</a></i>;<br/>


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