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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/>



<b>set </b><font color="Maroon">ic</font> =  <a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">c</font>;<br/>
<b>thus </b><a NAME="E1:57"/>
(  <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> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:57_1"/><i><font color="Green">E44</font></i>: 
 <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c</font> is <a href="waybel_0.html#V4">directed-sups-inheriting</a>
 ;<br/>

<b>let </b><font color="Maroon">D</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c</font></span>)</span>; <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="E2:57_1"/><i><font color="Green">E45</font></i>: 
( 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/>

<b>then reconsider </b><font color="Maroon">E</font> = <font color="Maroon">D</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">L</font> <b>by </b><i><a class="ref" href="yellow_2.html#T7">YELLOW_2:7</a></i>;<br/>
<a NAME="E4:57_1"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">c</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">D</font> <a href="hidden.html#R1">=</a> <font color="Maroon">E</font>
 
<b>by </b><i><a class="ref" href="waybel15.html#T14">WAYBEL15:14</a></i>;<br/>
<b>assume </b><a NAME="E5:57_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">D</font>, <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<b>thus </b><a NAME="E6:57_1"/>
 <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">c</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/>

<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#K3">inclusion</a> <font color="Maroon">c</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> <font color="Maroon">D</font>
<b>by </b><i><a class="ref" href="waybel34.html#T1" target="_self">Th1</a>, , , <a class="ref" href="waybel_0.html#T7">WAYBEL_0:7</a></i>
<br/>.= 
<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#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/><br/>


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

<b>assume </b><a NAME="E2:57"/><i><font color="Green">E47</font></i>: 
 <a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">c</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 ;<br/>

<b>let </b><font color="Maroon">X</font> be  <a href="waybel_0.html#V1">directed</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">c</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D4">WAYBEL_0:def 4</a><br/>

<b>assume </b><a NAME="E3:57"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">X</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/>

<b>assume </b><a NAME="E4:57"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 ;<br/>

<a NAME="E5:57"/>
(  <a href="waybel_1.html#K3">inclusion</a> <font color="Maroon">c</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">X</font> &amp;  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>, <a href="yellow_2.html#K1">Image</a> <font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="waybel34.html#T1" target="_self">Th1</a>, , <a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>
<b>then </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#K3">inclusion</a> <font color="Maroon">c</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> = 
<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#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">X</font>
<b>by </b><i><a class="ref" href="tmap_1.html#T91">TMAP_1:91</a></i>
;<br/>
<a NAME="E7:57"/><b>then </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#K3">inclusion</a> <font color="Maroon">c</font></span>)</span> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <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">c</font></span>)</span>
 
;<br/>
<b>hence </b><a NAME="E8:57"/>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<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">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel15.html#T14">WAYBEL15:14</a></i>;<br/>


</div>
