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

<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">k</font> be  <a href="waybel_1.html#V8">kernel</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<font color="Maroon">L</font>;<br/>



<b>set </b><font color="Maroon">ck</font> =  <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font>;<br/>
<a NAME="E1:51"/>
<span class="p1"><a href="waybel_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span>,<span class="p2">(<span class="default"><a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">k</font></span>)</span></span><a href="waybel_1.html#K1">]</a></span> is <a href="waybel_1.html#V3">Galois</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T42">WAYBEL_1:42</a></i>;<br/>
<a NAME="E2:51"/><b>then </b>
(  <a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">k</font> is <a href="waybel_1.html#V5">lower_adjoint</a> &amp;  <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font> is <a href="waybel_1.html#V4">upper_adjoint</a> )
 
<b>by </b><i><a class="ref" href="waybel_1.html#D11">WAYBEL_1:def 11</a>, <a class="ref" href="waybel_1.html#D12">WAYBEL_1:def 12</a></i>;<br/>
<a NAME="E3:51"/><b>then </b><i><font color="Green">E44</font></i>: 
(  <a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">k</font> is  <a href="waybel_0.html#V18">sups-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">k</font></span>)</span>,<font color="Maroon">L</font> &amp;  <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font> is <a href="waybel_0.html#V17">infs-preserving</a> )
 
<b>by </b><i><a class="ref" href="waybel_1.html#T13">WAYBEL_1:13</a>, <a class="ref" href="waybel_1.html#T14">WAYBEL_1:14</a></i>;<br/>
<a NAME="E4:51"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">k</font> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">k</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span> &amp;  <a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">k</font> <a href="funct_2.html#R2">=</a>  <a href="waybel34.html#K1" target="_self">LowerAdj</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span> )
 
<b>by </b><i>, <a class="ref" href="waybel_1.html#T35">WAYBEL_1:35</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:51_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">k</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 ;<br/><b>thus </b><a NAME="E2:51_1"/>
 <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a><br/>

<b>assume </b><a NAME="E1:51_1_1"/>
( not <font color="Maroon">D</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">D</font> is <a href="waybel_0.html#V1">directed</a> )
 ;<br/>

<a NAME="E2:51_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">k</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>
<b>assume </b><a NAME="E3:51_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</font>,<font color="Maroon">L</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<b>then </b><i><font color="Green">E48</font></i>:  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span> = 
<font color="Maroon">k</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel34.html#T1" target="_self">Th1</a>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i><a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
;<br/>
<b>thus </b><a NAME="E5:51_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font>, <a href="yellow_2.html#K1">Image</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<a NAME="E6:51_1_1"/><i><font color="Green">E88</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">k</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E7:51_1_1"/><i><font color="Green">E89</font></i>: 
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">k</font> is <a href="yellow_0.html#V8">sups-inheriting</a>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T56">WAYBEL_1:56</a></i>;<br/>
<a NAME="E8:51_1_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font>,<font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<a NAME="E9:51_1_1"/><b>then </b><i><font color="Green">E90</font></i>: 
 <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>,<font color="Maroon">L</font> <a href="hidden.html#R2">in</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">k</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="yellow_0.html#D19">YELLOW_0:def 19</a></i>;<br/>
<a NAME="E10:51_1_1"/>
<span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">k</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel15.html#T14">WAYBEL15:14</a></i>;<br/>
<b>hence </b> <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span> = 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">k</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="yellow_0.html#T65">YELLOW_0:65</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">D</font></span>)</span>
<b>by </b><i>, , <a class="ref" href="relat_1.html#T159">RELAT_1:159</a></i>
;<br/><br/>


</div><b>end;</b></div></div>
<b>end;</b></div>

<b>thus </b><a NAME="E6:51"/>
(  <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> implies <font color="Maroon">k</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> )
 <b>by </b><i>, , <a class="ref" href="waybel15.html#T13">WAYBEL15:13</a></i>;<br/>


</div>
