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

<b>let </b><font color="Maroon">S</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">T</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>;<br/><b>let </b><font color="Maroon">f</font> be  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="waybel_1.html#V6">projection</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T</font>,<font color="Maroon">T</font>;<br/>



<a NAME="E1:55"/>
 <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">IUPS</font> =  <a href="yellow_2.html#K1">Image</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f</font></span>)</span> as  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="waybel15.html#T1">WAYBEL15:1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">If</font> =  <a href="yellow_2.html#K1">Image</a> <font color="Maroon">f</font> as  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a> <b>by </b><i><a class="ref" href="yellow_2.html#T37">YELLOW_2:37</a></i>;<br/>
<a NAME="E4:55"/><i><font color="Green">E36</font></i>: 
 <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">If</font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">f</font></span>)</span> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:55"/>
<font color="Maroon">If</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> is  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font>
 
<b>by </b><i><a class="ref" href="yellow16.html#T41">YELLOW16:41</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">UPSIf</font> =  <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">If</font> as  <a href="yellow_0.html#V4">full</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">T</font> <a href="yellow_1.html#K6">|^</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> <b>by </b><i><a class="ref" href="waybel27.html#T1" target="_self">Th1</a>, <a class="ref" href="waybel15.html#T1">WAYBEL15:1</a></i>;<br/>
<a NAME="E7:55"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UPSIf</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IUPS</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:55_1"/>
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UPSIf</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IUPS</font>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:55_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UPSIf</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">h</font> = <font color="Maroon">x</font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">If</font> <b>by </b><i/>;<br/>
<a NAME="E3:55_1_1"/><i><font color="Green">E38</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">If</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D13">YELLOW_0:def 13</a></i>;<br/>
<a NAME="E4:55_1_1"/><i><font color="Green">E39</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">If</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:55_1_1"/>
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">h</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 
<b>by </b><i>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">h</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E7:55_1_1"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">g</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</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="E1:55_1_1_1"/><i><font color="Green">E51</font></i>: 
( not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">X</font> is <a href="waybel_0.html#V1">directed</a> )
 ;<br/>

<b>thus </b><a NAME="E2:55_1_1_1"/>
<font color="Maroon">g</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1_1_1_1"/><i><font color="Green">E54</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<b>reconsider </b><font color="Maroon">hX</font> = <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</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">If</font> <b>by </b><i>, , <a class="ref" href="relat_1.html#T152">RELAT_1:152</a>, <a class="ref" href="yellow_2.html#T17">YELLOW_2:17</a></i>;<br/>
<a NAME="E3:55_1_1_1_1"/>
<font color="Maroon">h</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#D37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E4:55_1_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">If</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">hX</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</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/>
<b>thus </b><a NAME="E5:55_1_1_1_1"/><i><font color="Green">E56</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">T</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<a NAME="E6:55_1_1_1_1"/>
<font color="Maroon">If</font> is  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="waybel_0.html#V4">directed-sups-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T37">YELLOW_2:37</a></i>;<br/>
<b>hence </b><a NAME="E7:55_1_1_1_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g</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#T7">WAYBEL_0:7</a></i>;<br/>


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


</div><b>end;</b></div>
<a NAME="E8:55_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<font color="Maroon">g</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="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
 
<b>by </b><i/>;<br/>
<a NAME="E9:55_1_1"/><i><font color="Green">E58</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<font color="Maroon">T</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">g</font>
<b>by </b><i><a class="ref" href="grcat_1.html#T12">GRCAT_1:12</a></i>
;<br/>
<a NAME="E11:55_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E12:55_1_1"/><b>then </b>
<font color="Maroon">x</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_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>hence </b><a NAME="E13:55_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IUPS</font>
 ;<br/>


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

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E2:55_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IUPS</font>
 ;<br/>

<a NAME="E3:55_1"/><b>then </b>
<font color="Maroon">x</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_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p3">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p4">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f</font></span>)</span></span>)</span></span>)</span>
 
;<br/>
<a NAME="E4:55_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f</font></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:55_1"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f</font></span>)</span> &amp; <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</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">a</font> = <font color="Maroon">a</font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> <b>by </b><i>, </i>;<br/>
<i><font color="Green">E72</font></i>: <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">a</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="grcat_1.html#T12">GRCAT_1:12</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon">x0</font> = <font color="Maroon">x</font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font> <b>by </b><i><a class="ref" href="waybel15.html#T13">WAYBEL15:13</a></i>;<br/>
<a NAME="E10:55_1"/><i><font color="Green">E73</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">x0</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">S</font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">x0</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">T</font> )
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E11:55_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</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_0.html#K5">subrelstr</a> <span class="p2">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="yellow_0.html#D15">YELLOW_0:def 15</a></i>;<br/>
<a NAME="E12:55_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">If</font>
 
;<br/>
<a NAME="E13:55_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">x0</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">If</font>
 
<b>by </b><i>, <a class="ref" href="relat_1.html#T45">RELAT_1:45</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">x1</font> = <font color="Maroon">x0</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S</font>,<font color="Maroon">If</font> <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E15:55_1"/>
<font color="Maroon">x1</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">X</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">S</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="E1:55_1_3"/><i><font color="Green">E74</font></i>: 
( not <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Maroon">X</font> is <a href="waybel_0.html#V1">directed</a> )
 ;<br/>

<b>thus </b><a NAME="E2:55_1_3"/>
<font color="Maroon">x1</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">X</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1_3_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:55_1_3_1"/><i><font color="Green">E97</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<b>reconsider </b><font color="Maroon">hX</font> = <font color="Maroon">x0</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</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">T</font> <b>by </b><i>, , <a class="ref" href="relat_1.html#T152">RELAT_1:152</a>, <a class="ref" href="yellow_2.html#T17">YELLOW_2:17</a></i>;<br/>
<a NAME="E3:55_1_3_1"/><i><font color="Green">E98</font></i>: 
<font color="Maroon">x0</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#D37">WAYBEL_0:def 37</a></i>;<br/>
<a NAME="E4:55_1_3_1"/><b>then </b><i><font color="Green">E99</font></i>: 
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">x0</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">T</font> &amp;  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">hX</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x0</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/>
<b>thus </b><a NAME="E5:55_1_3_1"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">x1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">If</font>
 <b>by </b><i><a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a></i>;<br/>

<a NAME="E6:55_1_3_1"/>
<font color="Maroon">x1</font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i/>;<br/>
<b>then reconsider </b><font color="Maroon">gX</font> = <font color="Maroon">x1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</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">If</font> <b>by </b><i>, , <a class="ref" href="relat_1.html#T152">RELAT_1:152</a>, <a class="ref" href="yellow_2.html#T17">YELLOW_2:17</a></i>;<br/>
<a NAME="E8:55_1_3_1"/>
<font color="Maroon">If</font> is  non <a href="struct_0.html#V3">empty</a> <a href="yellow_0.html#V4">full</a> <a href="waybel_0.html#V4">directed-sups-inheriting</a>  <a href="yellow_0.html#M1">SubRelStr</a> of <font color="Maroon">T</font>
 
<b>by </b><i><a class="ref" href="yellow_2.html#T37">YELLOW_2:37</a></i>;<br/>
<a NAME="E9:55_1_3_1"/><b>then </b>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">gX</font> <a href="hidden.html#R1">=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <font color="Maroon">hX</font>
 
<b>by </b><i>, <a class="ref" href="waybel_0.html#T7">WAYBEL_0:7</a></i>;<br/>
<b>hence </b><a NAME="E10:55_1_3_1"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">x1</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</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/>


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


</div><b>end;</b></div>
<b>hence </b><a NAME="E16:55_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">UPSIf</font>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:55"/>
 <a href="yellow_2.html#K1">Image</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S</font></span>)</span>,<font color="Maroon">f</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S</font>,<span class="p1">(<span class="default"><a href="yellow_2.html#K1">Image</a> <font color="Maroon">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#T58">YELLOW_0:58</a></i>;<br/>


</div>
