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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">  <a href="waybel19.html#K2" title="WAYBEL19:func.2">lambda</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="waybel28.html#K4" title="WAYBEL28:func.4">xi</a> <font color="Maroon" title="c1">L</font></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c2">T</font> being   <a href="yellow_9.html#NV1" title="YELLOW_9:NV.1">correct</a>   <a href="waybel19.html#V2" title="WAYBEL19:attr.2">Lawson</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">consider </span><font color="Maroon" title="c3">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">L</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c4">LL</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">L</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c5">LI</font> being   <a href="waybel33.html#V1" title="WAYBEL33:attr.1">lim-inf</a>   <a href="yellow_9.html#M1" title="YELLOW_9:mode.1">TopAugmentation</a> of <font color="Maroon" title="c1">L</font>;<br/>
<a NAME="E5:38"/><span class="lab"><font color="Green" title="E29">A1</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="c5">LI</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c5">LI</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">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</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/>
<a NAME="E6:38"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
 <a href="waybel28.html#K4" title="WAYBEL28:func.4">xi</a> <font color="Maroon" title="c1">L</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="c5">LI</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel33.html#T10" target="_self" title="WAYBEL33:th.10">Th10</a></span>;<br/>
<a NAME="E7:38"/>
 <a href="waybel19.html#K1" title="WAYBEL19:func.1">omega</a> <font color="Maroon" title="c1">L</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="c4">LL</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel19.html#D2" title="WAYBEL19:def.2">WAYBEL19:def 2</a></span>;<br/>
<a NAME="E8:38"/><span class="kw">then </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="c4">LL</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">LL</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">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> #) &amp; the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c4">LL</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="waybel33.html#T21" target="_self" title="WAYBEL33:th.21">Th21</a>, <a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
<a NAME="E9:38"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c5">LI</font> is    <a href="yellow_9.html#M2" title="YELLOW_9:mode.2">TopExtension</a> of <font color="Maroon" title="c4">LL</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:38"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E5:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="yellow_9.html#D5" title="YELLOW_9:def.5">YELLOW_9:def 5</a></span>;<br/>
<a NAME="E10:38"/>
 <a href="waybel11.html#K5" title="WAYBEL11:func.5">sigma</a> <font color="Maroon" title="c1">L</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">S</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="E11:38"/><span class="kw">then </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="c3">S</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">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">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c1">L</font> #) &amp; the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c3">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="waybel33.html#T19" target="_self" title="WAYBEL33:th.19">Th19</a>, <a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></span>;<br/>
<a NAME="E12:38"/><span class="kw">then </span>
( <font color="Maroon" title="c2">T</font> is    <a href="yellow_9.html#M3" title="YELLOW_9:mode.3">Refinement</a> of <font color="Maroon" title="c3">S</font>,<font color="Maroon" title="c4">LL</font> &amp; <font color="Maroon" title="c5">LI</font> is    <a href="yellow_9.html#M2" title="YELLOW_9:mode.2">TopExtension</a> of <font color="Maroon" title="c3">S</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:38"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E5:38"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="waybel19.html#T29" title="WAYBEL19:th.29">WAYBEL19:29</a>, <a class="ref" href="yellow_9.html#D5" title="YELLOW_9:def.5">YELLOW_9:def 5</a></span>;<br/>
<a NAME="E13:38"/><span class="kw">then </span>
(  <a href="waybel19.html#K2" title="WAYBEL19:func.2">lambda</a> <font color="Maroon" title="c1">L</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">T</font> &amp; <font color="Maroon" title="c5">LI</font> is    <a href="yellow_9.html#M2" title="YELLOW_9:mode.2">TopExtension</a> of <font color="Maroon" title="c2">T</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:38"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="waybel33.html#T22" target="_self" title="WAYBEL33:th.22">Th22</a>, <a class="ref" href="waybel19.html#D4" title="WAYBEL19:def.4">WAYBEL19:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E14:38"/>
 <a href="waybel19.html#K2" title="WAYBEL19:func.2">lambda</a> <font color="Maroon" title="c1">L</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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="txt" href="#E6:38"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="yellow_9.html#D5" title="YELLOW_9:def.5">YELLOW_9:def 5</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/>


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