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

<b>let </b><font color="Maroon">c<sub>1</sub></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<sub>2</sub></font> be  <a href="waybel_1.html#V7">closure</a> <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font>;<br/>



<a NAME="E1:58"/><i><font color="Green">E39</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<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="ref" href="waybel34.html#T39" target="_self">Th39</a>, <a class="ref" href="waybel_1.html#T32">WAYBEL_1:32</a></i>;<br/>
<a NAME="E2:58"/><i><font color="Green">E40</font></i>: 
(  <a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">c<sub>2</sub></font> is  <a href="waybel_0.html#V17">infs-preserving</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> &amp; (  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></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<sub>2</sub></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<sub>2</sub></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<sub>2</sub></font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a> ) )
 
<b>by </b><i><a class="ref" href="waybel34.html#T39" target="_self">Th39</a>, <a class="ref" href="waybel34.html#T40" target="_self">Th40</a></i>;<br/>
<b>hence </b><a NAME="E3:58"/>
(  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a> implies for <font color="Olive">b<sub>1</sub></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<sub>2</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> is <a href="t_0topsp.html#V1">open</a> )
 <b>by </b><i><a class="txt" href="waybel34.html#E1:58"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel34.html#T32" target="_self">Th32</a></i>;<br/>

<b>assume </b><a NAME="E4:58"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></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<sub>2</sub></font><br/>for <font color="Olive">b<sub>2</sub></font> being <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM7">Function</a> of <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> st <font color="Olive">b<sub>3</sub></font> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>3</sub></font> is <a href="t_0topsp.html#V1">open</a>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></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<sub>2</sub></font>, <font color="Maroon">c<sub>4</sub></font> being  <a href="waybel11.html#V4">Scott</a>  <a href="yellow_9.html#M1">TopAugmentation</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>3</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>3</sub></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<sub>2</sub></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<sub>2</sub></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">c<sub>4</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>4</sub></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">c<sub>1</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></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">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> <b>by </b><i><a class="txt" href="waybel34.html#E1:58"><i><font color="Green">E39</font></i></a></i>;<br/>
<a NAME="E8:58"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="t_0topsp.html#V1">open</a>
 
<b>by </b><i><a class="txt" href="waybel34.html#E4:58"><i><font color="Green">E41</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:58"/>
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c<sub>2</sub></font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a>
 <b>by </b><i><a class="txt" href="waybel34.html#E1:58"><i><font color="Green">E39</font></i></a>, <a class="txt" href="waybel34.html#E2:58"><i><font color="Green">E40</font></i></a>, <a class="ref" href="waybel34.html#T32" target="_self">Th32</a></i>;<br/>


</div>
