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

<b>let </b><font color="Maroon">R</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>, <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="struct_0.html#NM6">Function</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">R</font>,<font color="Maroon">S</font></span><a href="yellow_3.html#K3">:]</a></span>,<font color="Maroon">T</font>;<br/>



<b>assume </b><a NAME="E1:29"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">a</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><br/> for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> holds <br/> (  <a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Olive">a</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> &amp;  <a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Olive">b</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a> )
 ;<br/>

<a NAME="E2:29"/>
for <font color="Olive">X</font> being  <a href="struct_0.html#NM2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">R</font>,<font color="Maroon">S</font></span><a href="yellow_3.html#K3">:]</a></span>  st not <font color="Olive">X</font> is <a href="xboole_0.html#V1">empty</a> &amp; <font color="Olive">X</font> is <a href="waybel_0.html#V1">directed</a> holds <br/><font color="Maroon">f</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Olive">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_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 <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">R</font>,<font color="Maroon">S</font></span><a href="yellow_3.html#K3">:]</a></span>;<br/>

<b>assume </b><a NAME="E1:29_1"/>
( 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>then reconsider </b><font color="Maroon">X</font> = <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 <span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">R</font>,<font color="Maroon">S</font></span><a href="yellow_3.html#K3">:]</a></span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>;<br/><b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font>;<br/><a NAME="E1:29_1_1"/>
 <a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">a</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E2:29_1_1"/>
 <a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">a</font> is <a href="orders_3.html#V5">monotone</a>
 <b>by </b><i><a class="ref" href="waybel17.html#T3">WAYBEL17:3</a></i>;<br/><a NAME="E3:29_1_1"/>
 <a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E4:29_1_1"/>
 <a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b</font> is <a href="orders_3.html#V5">monotone</a>
 <b>by </b><i><a class="ref" href="waybel17.html#T3">WAYBEL17:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:29_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">f</font> is <a href="orders_3.html#V5">monotone</a>
 
<b>by </b><i/>;<br/>
<a NAME="E5:29_1"/>
<font color="Maroon">f</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="29_1_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:29_1_2"/><i><font color="Green">E44</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">X</font>,<span class="p1"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">R</font>,<font color="Maroon">S</font></span><a href="yellow_3.html#K3">:]</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a><br/>

<a NAME="E2:29_1_2"/><i><font color="Green">E53</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">f</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="E3:29_1_2"/><i><font color="Green">E58</font></i>: 
(  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font> is <a href="waybel_0.html#V1">directed</a> &amp; not  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T21">YELLOW_3:21</a>, <a class="ref" href="yellow_3.html#T22">YELLOW_3:22</a></i>;<br/>
<a NAME="E4:29_1_2"/><i><font color="Green">E59</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font>,<font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="yellow_3.html#T41">YELLOW_3:41</a></i>;<br/>
<a NAME="E5:29_1_2"/><i><font color="Green">E60</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a>  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font>,<font color="Maroon">S</font>
 
<b>by </b><i>, <a class="ref" href="yellow_3.html#T41">YELLOW_3:41</a></i>;<br/>
<a NAME="E6:29_1_2"/>
 <a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">S</font></span>)</span> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i/>;<br/>
<a NAME="E7:29_1_2"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">S</font></span>)</span> <a href="waybel_0.html#R4">preserves_sup_of</a>  <a href="yellow_3.html#K4">proj1</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/>
<i><font color="Green">E77</font></i>: <font color="Maroon">f</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> = 
<font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_3.html#T46">YELLOW_3:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span></span>)</span></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> <span class="p2">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<span class="p3">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p4">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
;<br/>
<a NAME="E9:29_1_2"/>
for <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>  st <font color="Olive">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font></span>)</span> holds <br/><font color="Olive">b</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">b</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>;<br/>

<b>assume </b><a NAME="E1:29_1_2_2"/>
<font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">b1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_1_2_2"/><i><font color="Green">E86</font></i>: 
( <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">b1</font> = <font color="Maroon">b1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> <b>by </b><i/>;<br/>
<a NAME="E5:29_1_2_2"/><i><font color="Green">E87</font></i>: 
( not  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font> is <a href="xboole_0.html#V1">empty</a> &amp;  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font> is <a href="waybel_0.html#V1">directed</a> )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T21">YELLOW_3:21</a>, <a class="ref" href="yellow_3.html#T22">YELLOW_3:22</a></i>;<br/>
<a NAME="E6:29_1_2_2"/>
 <a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b1</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 
<b>by </b><i/>;<br/>
<a NAME="E7:29_1_2_2"/><b>then </b><i><font color="Green">E88</font></i>: 
 <a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b1</font> <a href="waybel_0.html#R4">preserves_sup_of</a>  <a href="yellow_3.html#K5">proj2</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/>
<i><font color="Green">E89</font></i>: <font color="Maroon">b</font> = 
<span class="p1">(<span class="default"><a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b1</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> <span class="p2">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i>, </i>
<br/>.= 
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b1</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="waybel_0.html#D31">WAYBEL_0:def 31</a></i>
;<br/>
<a NAME="E9:29_1_2_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_2_2"><b>proof </b></a><div class="add">

<a NAME="E1:29_1_2_2_2"/><i><font color="Green">E90</font></i>: 
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b1</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span>,<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="E2:29_1_2_2_2"/>
<span class="p1">(<span class="default"><a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b1</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2_2_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:29_1_2_2_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b1</font></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">k1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:29_1_2_2_2_1"/><i><font color="Green">E94</font></i>: 
( <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b1</font></span>)</span> &amp; <font color="Maroon">k1</font> <a href="hidden.html#R2">in</a>  <a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font> &amp; <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K1" target="_self">Proj</a> <font color="Maroon">f</font>,<font color="Maroon">b1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">k1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">k1</font> = <font color="Maroon">k1</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">S</font> <b>by </b><i/>;<br/>
<a NAME="E5:29_1_2_2_2_1"/>
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">b1</font>,<font color="Maroon">k1</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:29_1_2_2_2_1"/>
<font color="Maroon">k</font> <a href="orders_2.html#R1">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, , , , </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1_2_2_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>
 <b>by </b><i>, , <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:29_1_2_2"/>
<font color="Maroon">b</font> <a href="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:29_1_2"/><b>then </b><i><font color="Green">E95</font></i>: 
<span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font></span>)</span> <a href="lattice3.html#R2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice3.html#D9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E11:29_1_2"/>
 <a href="yellow_0.html#R1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" target="_self">Proj</a> <font color="Maroon">f</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5">proj2</a> <font color="Maroon">X</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4">proj1</a> <font color="Maroon">X</font></span>)</span>,<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="E12:29_1_2"/><b>then </b><i><font color="Green">E96</font></i>: 
<font color="Maroon">f</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="orders_2.html#R3">&lt;=</a>  <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="yellow_0.html#D9">YELLOW_0:def 9</a></i>;<br/>
<a NAME="E13:29_1_2"/>
 <a href="yellow_0.html#NK3" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">f</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="waybel17.html#T16">WAYBEL17:16</a></i>;<br/>
<b>hence </b><a NAME="E14:29_1_2"/>
(  <a href="yellow_0.html#R1">ex_sup_of</a> <font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font>,<font color="Maroon">T</font> &amp;  <a href="yellow_0.html#K1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <font color="Maroon">X</font></span>)</span>,<font color="Maroon">T</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1">"\/"</a> <font color="Maroon">X</font>,<span class="p2"><a href="yellow_3.html#K3">[:</a><span class="default"><font color="Maroon">R</font>,<font color="Maroon">S</font></span><a href="yellow_3.html#K3">:]</a></span></span>)</span> )
 <b>by </b><i>, <a class="ref" href="yellow_0.html#T17">YELLOW_0:17</a>, <a class="ref" href="yellow_0.html#D3">YELLOW_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:29_1"/>
<font color="Maroon">f</font> <a href="waybel_0.html#R4">preserves_sup_of</a> <font color="Maroon">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:29"/>
<font color="Maroon">f</font> is <a href="waybel_0.html#V22">directed-sups-preserving</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D37">WAYBEL_0:def 37</a></i>;<br/>


</div>
