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<b>let </b><font color="Maroon" title="c1">T</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/><b>let </b><font color="Maroon" title="c2">S</font> be  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>



<b>thus </b><a NAME="E1:104"/>
( <font color="Maroon" title="c2">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> implies <font color="Maroon" title="c2">S</font> is <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:104_1"/><i><font color="Green" title="E70">A1</font></i>: 
<font color="Maroon" title="c2">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_6.html#D1" title="WAYBEL_6:def.1">WAYBEL_6:def 1</a><br/>

<b>assume </b><a NAME="E2:104_1"/>
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font>
 ;<br/>

<a NAME="E3:104_1"/><b>then </b>
 ex <font color="Olive" title="b1">y</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive" title="b1">y</font> <a href="waybel_3.html#NR2" title="WAYBEL_3:NR.2">&lt;&lt;</a> <font color="Maroon" title="c3">x</font> &amp; <font color="Olive" title="b1">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> )
 
<b>by </b><i><a class="txt" href="waybel11.html#E1:104_1"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="waybel11.html#T43" target="_self" title="WAYBEL11:th.43">Th43</a></i>;<br/>
<b>hence </b><a NAME="E4:104_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">S</font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="waybel_3.html#R1" title="WAYBEL_3:pred.1">is_way_below</a> <font color="Maroon" title="c3">x</font> )
 ;<br/>


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

<b>thus </b><a NAME="E2:104"/>
( <font color="Maroon" title="c2">S</font> is <a href="waybel_6.html#V1" title="WAYBEL_6:attr.1">Open</a> implies <font color="Maroon" title="c2">S</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 <b>by </b><i><a class="ref" href="waybel11.html#T41" target="_self" title="WAYBEL11:th.41">Th41</a></i>;<br/>


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