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<b>let </b><font color="Maroon">T</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V2">reflexive</a> <a href="orders_2.html#V3">transitive</a> <a href="waybel_0.html#V24">up-complete</a> <a href="waybel11.html#V4" target="_self">Scott</a>  <a href="waybel_9.html#L1">TopRelStr</a> ;<br/><b>let </b><font color="Maroon">S</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font>;<br/>



<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:28_1"/>
<font color="Maroon">S</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/><a NAME="E2:28_1"/><b>then </b>
<font color="Maroon">S</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/><b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">S</font> <a href="subset_1.html#K3">`</a>  as  <a href="waybel_0.html#V13">upper</a> <a href="waybel11.html#V1" target="_self">inaccessible</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> <b>by </b><i/>;<br/><a NAME="E4:28_1"/>
( <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="waybel11.html#V2" target="_self">closed_under_directed_sups</a> &amp; <font color="Maroon">A</font> <a href="subset_1.html#K3">`</a>  is <a href="waybel_0.html#V12">lower</a> )
 ;<br/><b>hence </b><a NAME="E5:28_1"/>
( <font color="Maroon">S</font> is <a href="waybel11.html#V2" target="_self">closed_under_directed_sups</a> &amp; <font color="Maroon">S</font> is <a href="waybel_0.html#V12">lower</a> )
 ;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:28"/>
( <font color="Maroon">S</font> is <a href="waybel11.html#V2" target="_self">closed_under_directed_sups</a> &amp; <font color="Maroon">S</font> is <a href="waybel_0.html#V12">lower</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">S</font> = <font color="Maroon">S</font> as  <a href="waybel_0.html#V12">lower</a> <a href="waybel11.html#V2" target="_self">directly_closed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">T</font> ;<br/>
<a NAME="E4:28"/>
<font color="Maroon">S</font> <a href="subset_1.html#K3">`</a>  is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E5:28"/>
<font color="Maroon">S</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29">TOPS_1:29</a></i>;<br/>


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