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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="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">A</font> being <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> holds <br/> ( <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> iff <font color="Olive" title="b1">A</font> is <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a> )</span><br/><b>let </b><font color="Maroon" title="c2">A</font> be  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> iff <font color="Maroon" title="c2">A</font> is <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a> )</span><br/>



<div><b>hereby </b> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a> implies <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )</span>
<div class="add"><b>assume </b><a NAME="E1:60_1"/>
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> is <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a></span><br/><a NAME="E2:60_1"/><b>then </b>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c3">mA</font> = <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as  <a href="waybel11.html#V3" title="WAYBEL11:attr.3">property(S)</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="waybel19.html#T36" target="_self" title="WAYBEL19:th.36">Th36</a></i>;<br/><a NAME="E4:60_1"/>
<font color="Maroon" title="c3">mA</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">A</font>
 ;<br/><b>hence </b><a NAME="E5:60_1"/>
<font color="Maroon" title="c2">A</font> is <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:60"/>
<font color="Maroon" title="c2">A</font> is <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a></span><br/>

<b>then reconsider </b><font color="Maroon" title="c3">dA</font> = <font color="Maroon" title="c2">A</font> as  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> <a href="waybel11.html#NV5" title="WAYBEL11:NV.5">directly_closed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font> ;<br/>
<b>consider </b><font color="Maroon" title="c4">S</font> being  <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a>  <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">T</font>;<br/>
<a NAME="E5:60"/><i><font color="Green" title="E40">A1</font></i>: 
 <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c4">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">S</font> #) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="orders_2.html#G1" title="ORDERS_2:aggr.1">RelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">T</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">T</font> #)
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">mA</font> = <font color="Maroon" title="c3">dA</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c4">S</font> ;<br/>
<a NAME="E7:60"/>
( <font color="Maroon" title="c5">mA</font> is <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> &amp; <font color="Maroon" title="c5">mA</font> is <a href="waybel11.html#V1" title="WAYBEL11:attr.1">inaccessible_by_directed_joins</a> )
 
<b>by </b><i><a class="txt" href="#E5:60"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="waybel_0.html#T25" title="WAYBEL_0:th.25">WAYBEL_0:25</a>, <a class="ref" href="yellow_9.html#T47" title="YELLOW_9:th.47">YELLOW_9:47</a></i>;<br/>
<a NAME="E8:60"/><b>then </b>
( <font color="Maroon" title="c5">mA</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c1">T</font> is    <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c4">S</font> )
 
<b>by </b><i><a class="txt" href="#E5:60"><i><font color="Green" title="E40">A1</font></i></a>, <a class="ref" href="waybel11.html#D4" title="WAYBEL11:def.4">WAYBEL11:def 4</a>, <a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E9:60"/><b>then </b>
<font color="Maroon" title="c3">dA</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="ref" href="waybel19.html#T37" target="_self" title="WAYBEL19:th.37">Th37</a></i>;<br/>
<b>hence </b><a NAME="E10:60"/>
<font color="Maroon" title="c2">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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