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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a>  <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">A</font> is  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a>  holds <br/>( <font color="Olive" title="b1">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K4" title="WAYBEL28:func.4">xi</a> <font color="Maroon" title="c1">L</font> 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/><span class="kw">let </span><font color="Maroon" title="c2">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> is  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a>  implies ( <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K4" title="WAYBEL28:func.4">xi</a> <font color="Maroon" title="c1">L</font> 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/>



<span class="kw">consider </span><font color="Maroon" title="c3">T</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">L</font>;<br/>
<span class="kw">assume </span><a NAME="E2:39"/><span class="lab"><font color="Green" title="E26">A1</font></span>: 
<font color="Maroon" title="c2">A</font> is  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K4" title="WAYBEL28:func.4">xi</a> <font color="Maroon" title="c1">L</font> 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/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c4">A1</font> = <font color="Maroon" title="c2">A</font> as   <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">L</font> ;<br/>
<a NAME="E4:39"/><span class="lab"><font color="Green" title="E27">A2</font></span>: 
 <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">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</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="c3">T</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">T</font> #)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">A9</font> = <font color="Maroon" title="c2">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T</font> ;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c6">A9</font> = <font color="Maroon" title="c5">A9</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">T</font> ;<br/>
<a NAME="E7:39"/><span class="lab"><font color="Green" title="E28">A3</font></span>: 
<font color="Maroon" title="c4">A1</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  is  <a href="waybel_0.html#V13" title="WAYBEL_0:attr.13">upper</a> 
 
;<br/>
<span class="kw">thus </span><a NAME="E8:39"/>
( <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K4" title="WAYBEL28:func.4">xi</a> <font color="Maroon" title="c1">L</font> implies <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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K4" title="WAYBEL28:func.4">xi</a> <font color="Maroon" title="c1">L</font> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:39_1"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K4" title="WAYBEL28:func.4">xi</a> <font color="Maroon" title="c1">L</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="E2:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">A9</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:39"><span class="lab"><font color="Green" title="E28">A3</font></span></a>, <a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="waybel28.html#T30" target="_self" title="WAYBEL28:th.30">Th30</a></span>;<br/>
<a NAME="E3:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">A9</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_9.html#T51" title="YELLOW_9:th.51">YELLOW_9:51</a></span>;<br/>
<a NAME="E4:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">A9</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E5:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">A9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></span>;<br/>
<a NAME="E6:39_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">A9</font> is  <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel11.html#T7" title="WAYBEL11:th.7">WAYBEL11:7</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:39_1"/>
<font color="Maroon" title="c2">A</font> is  <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="yellow12.html#T20" title="YELLOW12:th.20">YELLOW12:20</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E9:39"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K4" title="WAYBEL28:func.4">xi</a> <font color="Maroon" title="c1">L</font></span><br/>

<a NAME="E10:39"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A4</font></span>: 
<font color="Maroon" title="c6">A9</font> is  <a href="waybel11.html#V2" title="WAYBEL11:attr.2">closed_under_directed_sups</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="yellow12.html#T20" title="YELLOW12:th.20">YELLOW12:20</a></span>;<br/>
<a NAME="E11:39"/>
<font color="Maroon" title="c6">A9</font> is  <a href="waybel_0.html#V12" title="WAYBEL_0:attr.12">lower</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:39"><span class="lab"><font color="Green" title="E26">A1</font></span></a>, <a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="waybel_0.html#T25" title="WAYBEL_0:th.25">WAYBEL_0:25</a></span>;<br/>
<a NAME="E12:39"/><span class="kw">then </span>
<font color="Maroon" title="c6">A9</font> is  <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:39"><span class="lab"><font color="Green" title="E29">A4</font></span></a>, <a class="ref" href="waybel11.html#T7" title="WAYBEL11:th.7">WAYBEL11:7</a></span>;<br/>
<a NAME="E13:39"/><span class="kw">then </span>
<font color="Maroon" title="c6">A9</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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tops_1.html#T29" title="TOPS_1:th.29">TOPS_1:29</a></span>;<br/>
<a NAME="E14:39"/><span class="kw">then </span>
<font color="Maroon" title="c6">A9</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/>
<a NAME="E15:39"/><span class="kw">then </span>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:39"><span class="lab"><font color="Green" title="E27">A2</font></span></a>, <a class="ref" href="yellow_9.html#T51" title="YELLOW_9:th.51">YELLOW_9:51</a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:39"/>
<font color="Maroon" title="c2">A</font> <a href="subset_1.html#K3" title="SUBSET_1:func.3">`</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel28.html#K4" title="WAYBEL28:func.4">xi</a> <font color="Maroon" title="c1">L</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel28.html#T29" target="_self" title="WAYBEL28:th.29">Th29</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
