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

<b>let </b><font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_3.html#V3">continuous</a> <a href="waybel11.html#V4" target="_self">Scott</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/><b>let </b><font color="Maroon">S</font> be  <a href="waybel_0.html#V13">upper</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<b>thus </b><a NAME="E1:105"/>
( <font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a> implies <font color="Maroon">S</font> is <a href="waybel_6.html#V1">Open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="105_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:105_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_6.html#D1">WAYBEL_6:def 1</a><br/>

<b>assume </b><a NAME="E2:105_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font>
 ;<br/>

<a NAME="E3:105_1"/><b>then </b>
 ex <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">y</font> <a href="waybel_3.html#NR2" target="_self">&lt;&lt;</a> <font color="Maroon">x</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:105_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">S</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="waybel_3.html#R1">is_way_below</a> <font color="Maroon">x</font> )
 ;<br/>


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

<b>thus </b><a NAME="E2:105"/>
( <font color="Maroon">S</font> is <a href="waybel_6.html#V1">Open</a> implies <font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a> )
 <b>by </b><i><a class="ref" href="waybel11.html#T4" target="_self">Th4</a></i>;<br/>


</div>
