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

<b>let </b><font color="Maroon">L</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#V3">transitive</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">k</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">L</font>,<font color="Maroon">L</font>;<br/>



<b>assume </b><a NAME="E1:22"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">k</font> is <a href="waybel_0.html#V18">sups-preserving</a>
 ;<br/>

<b>let </b><font color="Maroon">X</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#D33">WAYBEL_0:def 33</a><br/>

<a NAME="E2:22"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">k</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">X</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D33">WAYBEL_0:def 33</a></i>;<br/>
<b>set </b><font color="Maroon">f</font> =  <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font>;<br/>
<a NAME="E3:22"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">k</font> <a href="funct_2.html#R1">=</a>  <a href="waybel_1.html#K2">corestr</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="waybel_1.html#T32">WAYBEL_1:32</a></i>;<br/>
<b>assume </b><a NAME="E4:22"/><i><font color="Green">E39</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</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/>

<a NAME="E5:22"/><b>then </b><i><font color="Green">E40</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">k</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">L</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>
<a NAME="E6:22"/><i><font color="Green">E41</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">X</font></span>)</span> <a href="hidden.html#R1">=</a> <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">X</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>
<a NAME="E7:22"/><i><font color="Green">E42</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E8:22"/><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">k</font> <a href="hidden.html#R1">=</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#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E9:22"/><i><font color="Green">E45</font></i>: 
<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">X</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">k</font>
 
<b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon">fX</font> = <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">X</font> as   <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">k</font></span>)</span> ;<br/>
<a NAME="E11:22"/>
 <a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">fX</font>,<font color="Maroon">L</font> is   <a href="struct_0.html#NM1">Element</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="waybel20.html#T1" target="_self">Th1</a>, , <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>;<br/>
<b>hence </b><a NAME="E12:22"/>
 <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">X</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#T65">YELLOW_0:65</a></i>;<br/>

<b>thus </b><a NAME="E13:22"/>
 <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">X</font></span>)</span> <a href="hidden.html#R1">=</a> <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">X</font></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="yellow_0.html#T65">YELLOW_0:65</a></i>;<br/>


</div>
