<?xml version="1.0"?>
<div class="add">

<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>;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>] <b>means </b>$1 <a href="waybel11.html#NR1" title="WAYBEL11:NR.1">has_the_property_(S)</a> ;<br/>
<b>consider </b><font color="Maroon" title="c2">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>, <font color="Maroon" title="c3">R</font> being  <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a> <a href="waybel19.html#V1" title="WAYBEL19:attr.1">lower</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="E2:54"/><i><font color="Green" title="E37">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="c2">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">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> #) &amp;  <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">R</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">R</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/>
<a NAME="E3:54"/><i><font color="Green" title="E38">A2</font></i>: 
 ex <font color="Olive" title="b1">K</font> being   <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c1">T</font> st <br/>for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">K</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">A</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>reconsider </b><font color="Maroon" title="c4">K</font> = <span class="p1">(<span class="default"><a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">T</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="waybel19.html#K1" title="WAYBEL19:func.1">omega</a> <font color="Maroon" title="c1">T</font></span>)</span> as    <a href="cantor_1.html#M2" title="CANTOR_1:mode.2">prebasis</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="waybel19.html#D3" target="_self" title="WAYBEL19:def.3">Def3</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">K</font>
;<br/>

<b>let </b><font color="Maroon" title="c5">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c6">AS</font> = <font color="Maroon" title="c5">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="waybel19.html#E2:54"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">AR</font> = <font color="Maroon" title="c5">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c3">R</font> <b>by </b><i><a class="txt" href="waybel19.html#E2:54"><i><font color="Green" title="E37">A1</font></i></a></i>;<br/>
<b>assume </b><a NAME="E4:54_1"/>
<font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">K</font>
 ;<br/>

<a NAME="E5:54_1"/><b>then </b>
( ( <font color="Maroon" title="c5">A</font> <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">T</font> &amp;  <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">S</font> ) or ( <font color="Maroon" title="c5">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel19.html#K1" title="WAYBEL19:func.1">omega</a> <font color="Maroon" title="c1">T</font> &amp;  <a href="waybel19.html#K1" title="WAYBEL19:func.1">omega</a> <font color="Maroon" title="c1">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">R</font> ) )
 
<b>by </b><i><a class="ref" href="waybel19.html#D2" target="_self" title="WAYBEL19:def.2">Def2</a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a>, <a class="ref" href="yellow_9.html#T51" title="YELLOW_9:th.51">YELLOW_9:51</a></i>;<br/>
<a NAME="E6:54_1"/><b>then </b>
( <font color="Maroon" title="c6">AS</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> or <font color="Maroon" title="c7">AR</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></i>;<br/>
<a NAME="E7:54_1"/><b>then </b>
( <font color="Maroon" title="c6">AS</font> <a href="waybel11.html#NR1" title="WAYBEL11:NR.1">has_the_property_(S)</a>  or <font color="Maroon" title="c7">AR</font> <a href="waybel11.html#NR1" title="WAYBEL11:NR.1">has_the_property_(S)</a>  )
 
<b>by </b><i><a class="ref" href="waybel19.html#T35" target="_self" title="WAYBEL19:th.35">Th35</a>, <a class="ref" href="waybel11.html#T15" title="WAYBEL11:th.15">WAYBEL11:15</a></i>;<br/>
<b>hence </b><a NAME="E8:54_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">A</font>]
 <b>by </b><i><a class="txt" href="waybel19.html#E2:54"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="yellow12.html#T19" title="YELLOW12:th.19">YELLOW12:19</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:54"/><i><font color="Green" title="E39">A3</font></i>: 
for <font color="Olive" title="b1">F</font> being  <a href="struct_0.html#NM3" title="STRUCT_0:NM.3">Subset-Family</a> of <font color="Maroon" title="c1">T</font>  st ( for <font color="Olive" title="b2">A</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">F</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">A</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="setfam_1.html#K5" title="SETFAM_1:func.5">union</a> <font color="Olive" title="b1">F</font>]
 
<b>by </b><i><a class="txt" href="waybel19.html#E41"><i><font color="Green" title="E32">Lm3</font></i></a></i>;<br/>
<a NAME="E5:54"/><i><font color="Green" title="E40">A4</font></i>: 
for <font color="Olive" title="b1">A1</font>, <font color="Olive" title="b2">A2</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">T</font>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A1</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">A2</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">A2</font>]
 
<b>by </b><i><a class="txt" href="waybel19.html#E42"><i><font color="Green" title="E33">Lm4</font></i></a></i>;<br/>
<a NAME="E6:54"/><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">T</font>]
 
;<br/>
<b>thus </b><a NAME="E7:54"/>
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">T</font>  st <font color="Olive" title="b1">A</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">A</font>]
 <b>from </b><i><a class="ref" href="waybel19.html#S1" target="_self" title="WAYBEL19:sch.1">WAYBEL19:sch 1</a>(<a class="txt" href="waybel19.html#E3:54"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="waybel19.html#E4:54"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="waybel19.html#E5:54"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="waybel19.html#E6:54"><i><font color="Green" title="E41">A5</font></i></a>);</i><br/>


</div>
