<?xml version="1.0"?>
<div class="add">

<b>let </b><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>;<br/><b>let </b><font color="Maroon" title="c2">c</font> be  <a href="waybel_1.html#V7" title="WAYBEL_1:attr.7">closure</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c1">L</font>;<br/>



<a NAME="E1:58"/><i><font color="Green" title="E38">A1</font></i>: 
(  <a href="waybel34.html#K1" title="WAYBEL34:func.1">LowerAdj</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">c</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">c</font> &amp;  <a href="waybel_1.html#K2" title="WAYBEL_1:func.2">corestr</a> <font color="Maroon" title="c2">c</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">c</font> )
 
<b>by </b><i><a class="ref" href="waybel34.html#T39" target="_self" title="WAYBEL34:th.39">Th39</a>, <a class="ref" href="waybel_1.html#T32" title="WAYBEL_1:th.32">WAYBEL_1:32</a></i>;<br/>
<a NAME="E2:58"/><i><font color="Green" title="E39">A2</font></i>: 
(  <a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">c</font> is  <a href="waybel_0.html#V17" title="WAYBEL_0:attr.17">infs-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span>,<font color="Maroon" title="c1">L</font> &amp; (  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font> is <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a> implies  <a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">c</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> ) &amp; (  <a href="waybel_1.html#K3" title="WAYBEL_1:func.3">inclusion</a> <font color="Maroon" title="c2">c</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> implies  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font> is <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a> ) )
 
<b>by </b><i><a class="ref" href="waybel34.html#T39" target="_self" title="WAYBEL34:th.39">Th39</a>, <a class="ref" href="waybel34.html#T40" target="_self" title="WAYBEL34:th.40">Th40</a></i>;<br/>
<b>hence </b><a NAME="E3:58"/>
(  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font> is <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a> implies for <font color="Olive" title="b1">X</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  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font><br/> for <font color="Olive" title="b2">Y</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/> for <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">Y</font>,<font color="Olive" title="b1">X</font>  st <font color="Olive" title="b3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">c</font> holds <br/><font color="Olive" title="b3">f</font> is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a> )
 <b>by </b><i><a class="txt" href="waybel34.html#E1:58"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="waybel34.html#T32" target="_self" title="WAYBEL34:th.32">Th32</a></i>;<br/>

<b>assume </b><a NAME="E4:58"/><i><font color="Green" title="E40">A3</font></i>: 
for <font color="Olive" title="b1">X</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  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font><br/> for <font color="Olive" title="b2">Y</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/> for <font color="Olive" title="b3">f</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Olive" title="b2">Y</font>,<font color="Olive" title="b1">X</font>  st <font color="Olive" title="b3">f</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">c</font> holds <br/><font color="Olive" title="b3">f</font> is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>
 ;<br/>

<b>consider </b><font color="Maroon" title="c3">X</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  <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font>, <font color="Maroon" title="c4">Y</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/>
<a NAME="E6:58"/>
(  <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">X</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c3">X</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 <span class="p1">(<span class="default"><a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</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="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font></span>)</span> #) &amp;  <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">Y</font>,the <a href="orders_2.html#U1" title="ORDERS_2:sel.1">InternalRel</a> of <font color="Maroon" title="c4">Y</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> #) )
 
<b>by </b><i><a class="ref" href="yellow_9.html#D4" title="YELLOW_9:def.4">YELLOW_9:def 4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">f</font> = <font color="Maroon" title="c2">c</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">Y</font>,<font color="Maroon" title="c3">X</font> <b>by </b><i><a class="txt" href="waybel34.html#E1:58"><i><font color="Green" title="E38">A1</font></i></a></i>;<br/>
<a NAME="E8:58"/>
<font color="Maroon" title="c5">f</font> is <a href="t_0topsp.html#V1" title="T_0TOPSP:attr.1">open</a>
 
<b>by </b><i><a class="txt" href="waybel34.html#E4:58"><i><font color="Green" title="E40">A3</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:58"/>
 <a href="yellow_2.html#K1" title="YELLOW_2:func.1">Image</a> <font color="Maroon" title="c2">c</font> is <a href="waybel_0.html#V4" title="WAYBEL_0:attr.4">directed-sups-inheriting</a>
 <b>by </b><i><a class="txt" href="waybel34.html#E1:58"><i><font color="Green" title="E38">A1</font></i></a>, <a class="txt" href="waybel34.html#E2:58"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="waybel34.html#T32" target="_self" title="WAYBEL34:th.32">Th32</a></i>;<br/>


</div>
