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<b>let </b><font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel19.html#V2" target="_self">Lawson</a> <a href="waybel_9.html#NM1">TopLattice</a>;<br/>

<b>consider </b><font color="Maroon">S</font> being  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">T</font>, <font color="Maroon">R</font> being  <a href="pre_topc.html#V2">correct</a> <a href="waybel19.html#V1" target="_self">lower</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">T</font>;<br/>
<a NAME="E2:56"/><i><font color="Green">E30</font></i>: 
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">S</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">R</font> #) <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">T</font> #) )
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<a NAME="E3:56"/>
<font color="Maroon">T</font> is    <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T44">YELLOW_9:44</a></i>;<br/>
<a NAME="E4:56"/><b>then </b><i><font color="Green">E34</font></i>: 
<font color="Maroon">T</font> is    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">S</font>,<font color="Maroon">R</font>
 
<b>by </b><i><a class="txt" href="waybel19.html#E2"><i><font color="Green">Lemma28</font></i></a></i>;<br/>
<a NAME="E5:56"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">T</font> is    <a href="yellow_9.html#M3">Refinement</a> of <font color="Maroon">R</font>,<font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T55">YELLOW_9:55</a></i>;<br/>
<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>

<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">z</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E8:56"/>
(  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> &amp;  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">y</font> is <a href="pre_topc.html#V4">closed</a> &amp;  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a>  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font> &amp;  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">z</font> is <a href="pre_topc.html#V4">closed</a> )
 
<b>by </b><i>, , , <a class="ref" href="waybel11.html#T11">WAYBEL11:11</a></i>;<br/>
<b>hence </b><a NAME="E9:56"/>
(  <a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font> is <a href="pre_topc.html#V4">closed</a> &amp;  <a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i>, , <a class="ref" href="waybel19.html#T1" target="_self">Th1</a>, </i>;<br/>

<a NAME="E10:56"/><b>then </b>
<span class="p1">(<span class="default"><a href="waybel_0.html#K6">uparrow</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="waybel_0.html#K5">downarrow</a> <font color="Maroon">x</font></span>)</span> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>
<b>hence </b><a NAME="E11:56"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">x</font></span><a href="domain_1.html#K6">}</a></span> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="ref" href="waybel19.html#D1" target="_self">Def1</a></i>;<br/>


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