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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>;<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="E2:56"/><i><font color="Green" title="E38">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="E3:56"/>
<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="c1">T</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T44" title="YELLOW_9:th.44">YELLOW_9:44</a></i>;<br/>
<a NAME="E4:56"/><b>then </b><i><font color="Green" title="E39">A2</font></i>: 
<font color="Maroon" title="c1">T</font> is    <a href="yellow_9.html#M3" title="YELLOW_9:mode.3">Refinement</a> of <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">R</font>
 
<b>by </b><i><a class="ref" href="waybel19.html#T29" target="_self" title="WAYBEL19:th.29">Th29</a></i>;<br/>
<a NAME="E5:56"/><b>then </b><i><font color="Green" title="E40">A3</font></i>: 
<font color="Maroon" title="c1">T</font> is    <a href="yellow_9.html#M3" title="YELLOW_9:mode.3">Refinement</a> of <font color="Maroon" title="c3">R</font>,<font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T55" title="YELLOW_9:th.55">YELLOW_9:55</a></i>;<br/>
<b>let </b><font color="Maroon" title="c4">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">T</font>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">y</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="waybel19.html#E2:56"><i><font color="Green" title="E38">A1</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">z</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">R</font> <b>by </b><i><a class="txt" href="waybel19.html#E2:56"><i><font color="Green" title="E38">A1</font></i></a></i>;<br/>
<a NAME="E8:56"/>
(  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">x</font> &amp;  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c5">y</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp;  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">z</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c4">x</font> &amp;  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c6">z</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 
<b>by </b><i><a class="txt" href="waybel19.html#E2:56"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="waybel19.html#T4" target="_self" title="WAYBEL19:th.4">Th4</a>, <a class="ref" href="waybel11.html#T11" title="WAYBEL11:th.11">WAYBEL11:11</a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></i>;<br/>
<b>hence </b><a NAME="E9:56"/>
(  <a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c4">x</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> &amp;  <a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">x</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <b>by </b><i><a class="txt" href="waybel19.html#E2:56"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="waybel19.html#E4:56"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="waybel19.html#E5:56"><i><font color="Green" title="E40">A3</font></i></a>, <a class="ref" href="waybel19.html#T21" target="_self" title="WAYBEL19:th.21">Th21</a></i>;<br/>

<a NAME="E10:56"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_0.html#K6" title="WAYBEL_0:func.6">uparrow</a> <font color="Maroon" title="c4">x</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5" title="WAYBEL_0:func.5">downarrow</a> <font color="Maroon" title="c4">x</font></span>)</span> 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#T35" title="TOPS_1:th.35">TOPS_1:35</a></i>;<br/>
<b>hence </b><a NAME="E11:56"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="ref" href="waybel19.html#T28" target="_self" title="WAYBEL19:th.28">Th28</a></i>;<br/>


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