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<span class="kw">let </span><font color="Maroon" title="c1">N</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_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a>  <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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">N</font><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">N</font>  st <font color="Olive" title="b2">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel19.html#K2" title="WAYBEL19:func.2">lambda</a> <font color="Maroon" title="c1">N</font> holds <br/> <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Olive" title="b2">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="Olive" title="b1">S</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <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">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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">N</font>  st <font color="Olive" title="b1">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel19.html#K2" title="WAYBEL19:func.2">lambda</a> <font color="Maroon" title="c1">N</font> holds <br/> <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Olive" title="b1">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="c2">S</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">N</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel19.html#K2" title="WAYBEL19:func.2">lambda</a> <font color="Maroon" title="c1">N</font> implies  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c3">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="c2">S</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:28"/>
<font color="Maroon" title="c3">A</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="waybel19.html#K2" title="WAYBEL19:func.2">lambda</a> <font color="Maroon" title="c1">N</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c3">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="c2">S</font></span><br/>

<a NAME="E2:28"/><span class="kw">then </span><span class="lab"><font color="Green" title="E13">A1</font></span>: 
<font color="Maroon" title="c3">A</font> 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="waybel30.html#T12" target="_self" title="WAYBEL30:th.12">Th12</a></span>;<br/>
<a NAME="E3:28"/><span class="lab"><font color="Green" title="E14">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">N</font>, the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">N</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="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> #)
 
<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="c4">Y</font> = <font color="Maroon" title="c3">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<a NAME="E5:28"/><span class="lab"><font color="Green" title="E15">A3</font></span>: 
<font color="Maroon" title="c2">S</font> is  <a href="waybel_2.html#V2" title="WAYBEL_2:attr.2">meet-continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="yellow12.html#T14" title="YELLOW12:th.14">YELLOW12:14</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c5">UA</font> =  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c3">A</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E14">A2</font></span></a></span>;<br/>
<a NAME="E7:28"/><span class="lab"><font color="Green" title="E16">A4</font></span>: 
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c3">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c4">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="waybel_0.html#T13" title="WAYBEL_0:th.13">WAYBEL_0:13</a></span>;<br/>
<a NAME="E8:28"/>
<font color="Maroon" title="c4">Y</font> is  <a href="waybel11.html#V3" title="WAYBEL11:attr.3">property(S)</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E13">A1</font></span></a>, <a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E14">A2</font></span></a>, <a class="ref" href="waybel19.html#T36" title="WAYBEL19:th.36">WAYBEL19:36</a>, <a class="ref" href="yellow12.html#T19" title="YELLOW12:th.19">YELLOW12:19</a></span>;<br/>
<a NAME="E9:28"/><span class="kw">then </span>
<font color="Maroon" title="c5">UA</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:28"><span class="lab"><font color="Green" title="E15">A3</font></span></a>, <a class="txt" href="#E7:28"><span class="lab"><font color="Green" title="E16">A4</font></span></a>, <a class="ref" href="waybel11.html#T15" title="WAYBEL11:th.15">WAYBEL11:15</a></span>;<br/>
<a NAME="E10:28"/><span class="kw">then </span>
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c3">A</font> <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="c2">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D2" title="PRE_TOPC:def.2">PRE_TOPC:def 2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:28"/>
 <a href="waybel_0.html#K4" title="WAYBEL_0:func.4">uparrow</a> <font color="Maroon" title="c3">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="c2">S</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel14.html#T23" title="WAYBEL14:th.23">WAYBEL14:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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