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<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="E1:38"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">L</font> is <a href="waybel_3.html#V3">continuous</a>
 ;<br/>

<a NAME="E2:38"/><i><font color="Green">E37</font></i>: 
 <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:38"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>  st <font color="Olive">X</font> is <a href="pre_topc.html#V3">open</a> holds <br/><font color="Olive">X</font> is <a href="waybel_0.html#V13">upper</a>
 
<b>by </b><i><a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<a NAME="E4:38"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="waybel_0.html#K6">uparrow</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K3">Bottom</a> <font color="Maroon">L</font></span>)</span> is <a href="compts_1.html#V6">compact</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:38"/>
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">L</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
;<br/>
<b>hence </b><a NAME="E6:38"/>
<font color="Maroon">L</font> is <a href="compts_1.html#V2">compact</a>
 <b>by </b><i>, <a class="ref" href="waybel14.html#T1" target="_self">Th1</a>, <a class="ref" href="compts_1.html#T10">COMPTS_1:10</a></i>;<br/>

<b>thus </b><a NAME="E7:38"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">L</font> is <a href="waybel_3.html#V6">locally-compact</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="38_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_3.html#D9">WAYBEL_3:def 9</a><br/><b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:38_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 <b>and </b><br/><a NAME="E2:38_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">X</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/>
<b>set </b><font color="Maroon">bas</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Olive">q</font></span>)</span> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">q</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x'</font> </span>}</span> ;<br/>
<a NAME="E4:38_1"/><i><font color="Green">E45</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Olive">q</font></span>)</span> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">q</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x'</font> </span>}</span>  is    <a href="yellow_8.html#M1">Basis</a> of <font color="Maroon">x</font>
 
<b>by </b><i>, <a class="ref" href="waybel11.html#T44">WAYBEL11:44</a></i>;<br/>
<b>consider </b><font color="Maroon">y</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font><b> such that </b><br/><a NAME="E6:38_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">y</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x'</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> )
 <b>by </b><i>, , , <a class="ref" href="waybel11.html#T43">WAYBEL11:43</a></i>;<br/>
<a NAME="E7:38_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">X</font> is <a href="waybel_0.html#V13">upper</a>
 
<b>by </b><i>, <a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a></i>;<br/>
<b>set </b><font color="Maroon">Y</font> =  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">y</font>;<br/>
<b>take </b>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">y</font>
;<br/>

<a NAME="E8:38_1"/>
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="waybel_3.html#K2">wayabove</a> <font color="Olive">q</font></span>)</span> where <font color="Olive">q</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> : <font color="Olive">q</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x'</font> </span>}</span> 
 
<b>by </b><i/>;<br/>
<a NAME="E9:38_1"/><b>then </b><i><font color="Green">E48</font></i>: 
(  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">y</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">y</font> )
 
<b>by </b><i>, <a class="ref" href="yellow_8.html#T21">YELLOW_8:21</a></i>;<br/>
<a NAME="E10:38_1"/>
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">y</font>
 
<b>by </b><i><a class="ref" href="waybel_3.html#T11">WAYBEL_3:11</a></i>;<br/>
<a NAME="E11:38_1"/><b>then </b>
 <a href="waybel_3.html#K2">wayabove</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">y</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel14.html#E2:38"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tops_1.html#T56">TOPS_1:56</a></i>;<br/>
<b>hence </b><a NAME="E12:38_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="tops_1.html#K1">Int</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">y</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel14.html#E2:38"><i><font color="Green">E37</font></i></a></i>;<br/>

<b>thus </b><a NAME="E13:38_1"/>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">y</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">X</font>
 <b>by </b><i>, <a class="txt" href="waybel14.html#E1:38"><i><font color="Green">E36</font></i></a>, <a class="ref" href="waybel11.html#T42">WAYBEL11:42</a></i>;<br/>

<b>thus </b><a NAME="E14:38_1"/>
 <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">y</font> is <a href="compts_1.html#V6">compact</a>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>

<a NAME="E8:38"/>
 <a href="waybel_2.html#K5">sup_op</a> <font color="Maroon">L</font> is <a href="waybel14.html#V1" target="_self">jointly_Scott-continuous</a>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E9:38"/>
<font color="Maroon">L</font> is <a href="yellow_8.html#V3">sober</a>
 <b>by </b><i/>;<br/>

<b>hence </b><a NAME="E10:38"/>
<font color="Maroon">L</font> is <a href="yellow_8.html#V1">Baire</a>
 <b>by </b><i>, <a class="ref" href="waybel12.html#T48">WAYBEL12:48</a></i>;<br/>


</div>
