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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a>;<br/><b>let </b><font color="Maroon" title="c2">L</font> be  non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel11.html#V4" title="WAYBEL11:attr.4">Scott</a> <a href="waybel_9.html#NM1" title="WAYBEL_9:NM.1">TopLattice</a>;<br/>



<a NAME="E1:50"/><i><font color="Green" title="E39">A1</font></i>: 
<font color="Maroon" title="c2">L</font> is <a href="waybel25.html#V1" title="WAYBEL25:attr.1">monotone-convergence</a>
 
<b>by </b><i><a class="ref" href="waybel25.html#T29" title="WAYBEL25:th.29">WAYBEL25:29</a></i>;<br/>
<a NAME="E2:50"/>
 <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">L</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="waybel_9.html#G1" title="WAYBEL_9:aggr.1">TopRelStr</a>(# the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">L</font>,the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c2">L</font> #)
 
<b>by </b><i><a class="ref" href="waybel25.html#T15" title="WAYBEL25:th.15">WAYBEL25:15</a></i>;<br/>
<a NAME="E3:50"/><b>then </b><i><font color="Green" title="E40">A2</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 <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">L</font></span>)</span>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <span class="p1">(<span class="default"><a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">L</font></span>)</span> #) <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">L</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c2">L</font> #)
 
;<br/>
<a NAME="E4:50"/><i><font color="Green" title="E41">A3</font></i>: 
<font color="Maroon" title="c2">L</font> is  <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="c2">L</font>
 
<b>by </b><i><a class="ref" href="yellow_9.html#T44" title="YELLOW_9:th.44">YELLOW_9:44</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:50_1"/><i><font color="Green" title="E42">A4</font></i>: 
(  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> &amp;  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> )
 ;<br/><b>hence </b><a NAME="E2:50_1"/>
 <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel26.html#E1:50"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="waybel26.html#T27" target="_self" title="WAYBEL26:th.27">Th27</a></i>;<br/><a NAME="E3:50_1"/>
 <a href="waybel25.html#K1" title="WAYBEL25:func.1">Omega</a> <font color="Maroon" title="c2">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel26.html#E4:50"><i><font color="Green" title="E41">A3</font></i></a>, <a class="txt" href="waybel26.html#E1:50_1"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="waybel26.html#T29" target="_self" title="WAYBEL26:th.29">Th29</a></i>;<br/><b>hence </b><a NAME="E4:50_1"/>
<font color="Maroon" title="c2">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a>
 <b>by </b><i><a class="txt" href="waybel26.html#E3:50"><i><font color="Green" title="E40">A2</font></i></a>, <a class="ref" href="yellow12.html#T15" title="YELLOW12:th.15">YELLOW12:15</a></i>;<br/></div>
<b>end;</b></div>

<a NAME="E6:50"/>
( <font color="Maroon" title="c2">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> implies <font color="Maroon" title="c2">L</font> is <a href="waybel18.html#V2" title="WAYBEL18:attr.2">injective</a> )
 
<b>by </b><i><a class="txt" href="waybel26.html#E4:50"><i><font color="Green" title="E41">A3</font></i></a>, <a class="ref" href="waybel25.html#T12" title="WAYBEL25:th.12">WAYBEL25:12</a></i>;<br/>
<b>hence </b><a NAME="E7:50"/>
(  <a href="yellow_1.html#K2" title="YELLOW_1:func.2">InclPoset</a> the <a href="pre_topc.html#U1" title="PRE_TOPC:sel.1">topology</a> of <font color="Maroon" title="c1">X</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> &amp; <font color="Maroon" title="c2">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> implies (  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">L</font> is <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> &amp;  <a href="waybel26.html#K1" title="WAYBEL26:func.1">oContMaps</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">L</font> is <a href="waybel_3.html#V3" title="WAYBEL_3:attr.3">continuous</a> ) )
 <b>by </b><i><a class="ref" href="waybel26.html#T37" target="_self" title="WAYBEL26:th.37">Th37</a></i>;<br/>


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