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<div class="add">

<b>let </b><font color="Maroon">R</font> be  <a href="waybel_0.html#V24">up-complete</a> <a href="waybel_0.html#V25">/\-complete</a> <a href="waybel_0.html#NM6">Semilattice</a>;<br/><b>let </b><font color="Maroon">T</font> be    <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">R</font>;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green">E32</font></i>: 
the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font> <a href="hidden.html#R1">=</a>  <a href="waybel11.html#K5">sigma</a> <font color="Maroon">R</font>
 ;<br/>

<a NAME="E2:18"/><i><font color="Green">E36</font></i>: 
 <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</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">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E3:18"/>
<font color="Maroon">T</font> is <a href="waybel11.html#V4">Scott</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D4">WAYBEL11:def 4</a><br/>

<b>reconsider </b><font color="Maroon">A</font> = <font color="Maroon">S</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<b>thus </b><a NAME="E2:18_1"/>
( <font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a> implies ( <font color="Maroon">S</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon">S</font> is <a href="waybel_0.html#V13">upper</a> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:18_1_1"/>
<font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a>
 ;<br/>

<a NAME="E2:18_1_1"/><b>then </b>
<font color="Maroon">S</font> <a href="hidden.html#R2">in</a> the <a href="pre_topc.html#U1">topology</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E3:18_1_1"/><b>then </b>
( <font color="Maroon">A</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon">A</font> is <a href="waybel_0.html#V13">upper</a> )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:18_1_1"/>
( <font color="Maroon">S</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon">S</font> is <a href="waybel_0.html#V13">upper</a> )
 <b>by </b><i>, <a class="ref" href="waybel_0.html#T25">WAYBEL_0:25</a>, <a class="ref" href="yellow_9.html#T47">YELLOW_9:47</a></i>;<br/>


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

<b>assume </b><a NAME="E3:18_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">S</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon">S</font> is <a href="waybel_0.html#V13">upper</a> )
 ;<br/>

<a NAME="E4:18_1"/>
<font color="Maroon">A</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="18_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D</font> be  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel11.html#D1">WAYBEL11:def 1</a><br/>

<b>assume </b><a NAME="E1:18_1_2"/><i><font color="Green">E39</font></i>: 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font>
 ;<br/>

<b>reconsider </b><font color="Maroon">E</font> = <font color="Maroon">D</font> as   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/>
<a NAME="E3:18_1_2"/>
 ex <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">a</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">D</font> &amp; ( for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>  st <font color="Olive">b</font> <a href="lattice3.html#NR4" target="_self">is_&gt;=_than</a> <font color="Maroon">D</font> holds <br/><font color="Olive">a</font> <a href="orders_2.html#R3">&lt;=</a> <font color="Olive">b</font> ) )
 
<b>by </b><i><a class="ref" href="waybel_0.html#D39">WAYBEL_0:def 39</a></i>;<br/>
<a NAME="E4:18_1_2"/><b>then </b>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</font>,<font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T15">YELLOW_0:15</a></i>;<br/>
<a NAME="E5:18_1_2"/><b>then </b>
( <font color="Maroon">E</font> is <a href="waybel_0.html#V1">directed</a> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">E</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T3">WAYBEL_0:3</a>, <a class="ref" href="yellow_0.html#T26">YELLOW_0:26</a></i>;<br/>
<b>hence </b><a NAME="E6:18_1_2"/>
<font color="Maroon">D</font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">A</font>
 <b>by </b><i>, , <a class="ref" href="waybel11.html#D1">WAYBEL11:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:18_1"/><b>then </b>
( <font color="Maroon">A</font> is <a href="waybel11.html#V1">inaccessible_by_directed_joins</a> &amp; <font color="Maroon">A</font> is <a href="waybel_0.html#V13">upper</a> )
 
<b>by </b><i>, , <a class="ref" href="waybel_0.html#T25">WAYBEL_0:25</a></i>;<br/>
<a NAME="E6:18_1"/><b>then </b>
<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">T</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E7:18_1"/>
<font color="Maroon">S</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D5">PRE_TOPC:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:18"/>
<font color="Maroon">T</font> is <a href="waybel11.html#V4">Scott</a>
 ;<br/>


</div>
