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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>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">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">N</font>;<br/>





<b>assume </b><a NAME="E1:28"/><i><font color="Green">E32</font></i>: 
<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/>

<a NAME="E2:28"/><i><font color="Green">E34</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>then reconsider </b><font color="Maroon">Y</font> = <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> ;<br/>
<b>reconsider </b><font color="Maroon">UA</font> =  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">A</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E5:28"/><i><font color="Green">E35</font></i>: 
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">A</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">Y</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T13">WAYBEL_0:13</a></i>;<br/>
<a NAME="E6:28"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">S</font> is <a href="waybel_2.html#V2">meet-continuous</a>
 
<b>by </b><i>, <a class="ref" href="yellow12.html#T14">YELLOW12:14</a></i>;<br/>
<a NAME="E7:28"/>
<font color="Maroon">A</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E8:28"/><b>then </b>
<font color="Maroon">A</font> is <a href="waybel11.html#V3">property(S)</a>
 
<b>by </b><i><a class="ref" href="waybel19.html#T36">WAYBEL19:36</a></i>;<br/>
<a NAME="E9:28"/><b>then </b>
<font color="Maroon">Y</font> is <a href="waybel11.html#V3">property(S)</a>
 
<b>by </b><i>, <a class="ref" href="yellow12.html#T19">YELLOW12:19</a></i>;<br/>
<a NAME="E10:28"/><b>then </b>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">Y</font> is <a href="waybel11.html#V3">property(S)</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:28"/><b>then </b>
<font color="Maroon">UA</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i>, <a class="ref" href="waybel11.html#T15">WAYBEL11:15</a></i>;<br/>
<a NAME="E12:28"/><b>then </b>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">A</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<b>hence </b><a NAME="E13:28"/>
 <a href="waybel_0.html#K4">uparrow</a> <font color="Maroon">A</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#T23">WAYBEL14:23</a></i>;<br/>


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