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<b>let </b><font color="Maroon">L</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">c</font> be  <a href="waybel_1.html#V7">closure</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<font color="Maroon">L</font>;<br/>



<a NAME="E1:58"/><i><font color="Green">E44</font></i>: 
(  <a href="waybel34.html#K1" target="_self">LowerAdj</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">c</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c</font> &amp;  <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c</font> )
 
<b>by </b><i>, <a class="ref" href="waybel_1.html#T32">WAYBEL_1:32</a></i>;<br/>
<a NAME="E2:58"/><i><font color="Green">E45</font></i>: 
(  <a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">c</font> is  <a href="waybel_0.html#V17">infs-preserving</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c</font></span>)</span>,<font color="Maroon">L</font> &amp; (  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c</font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a> implies  <a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">c</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> ) &amp; (  <a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">c</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> implies  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c</font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a> ) )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E3:58"/>
(  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c</font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a> implies for <font color="Olive">X</font> being <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c</font><br/> for <font color="Olive">Y</font> being <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">L</font><br/> for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Olive">Y</font>,<font color="Olive">X</font>  st <font color="Olive">f</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c</font> holds <br/><font color="Olive">f</font> is <a href="t_0topsp.html#V1">open</a> )
 <b>by </b><i><a class="ref" href="waybel34.html#T1" target="_self">Th1</a>, </i>;<br/>

<b>assume </b><a NAME="E4:58"/><i><font color="Green">E46</font></i>: 
for <font color="Olive">X</font> being <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c</font><br/> for <font color="Olive">Y</font> being <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">L</font><br/> for <font color="Olive">f</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Olive">Y</font>,<font color="Olive">X</font>  st <font color="Olive">f</font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c</font> holds <br/><font color="Olive">f</font> is <a href="t_0topsp.html#V1">open</a>
 ;<br/>

<b>consider </b><font color="Maroon">X</font> being  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c</font>, <font color="Maroon">Y</font> being  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">L</font>;<br/>
<a NAME="E6:58"/>
(  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">X</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 <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c</font></span>)</span>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c</font></span>)</span> #) &amp;  <a href="orders_2.html#G1">RelStr</a>(# the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">Y</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">L</font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">L</font> #) )
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4">YELLOW_9:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">c</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">Y</font>,<font color="Maroon">X</font> <b>by </b><i><a class="ref" href="waybel34.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E8:58"/>
<font color="Maroon">f</font> is <a href="t_0topsp.html#V1">open</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E9:58"/>
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c</font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a>
 <b>by </b><i><a class="ref" href="waybel34.html#T1" target="_self">Th1</a>, , </i>;<br/>


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