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

<b>let </b><font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c2">S</font>, <font color="Maroon" title="c3">T</font> be  <a href="lattice3.html#V3" title="LATTICE3:attr.3">complete</a> <a href="waybel_0.html#NM8" title="WAYBEL_0:NM.8">LATTICE</a>;<br/><b>let </b><font color="Maroon" title="c4">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>,<font color="Maroon" title="c3">T</font>;<br/>



<b>assume </b><a NAME="E1:29"/><i><font color="Green" title="E12">A1</font></i>: 
for <font color="Olive" title="b1">a</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><br/> for <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> holds <br/> (  <a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b1">a</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> &amp;  <a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Olive" title="b2">b</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> )
 ;<br/>

<a NAME="E2:29"/>
for <font color="Olive" title="b1">X</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>  st not <font color="Olive" title="b1">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Olive" title="b1">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> holds <br/><font color="Maroon" title="c4">f</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Olive" title="b1">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" title="c5">X</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>;<br/>

<b>assume </b><a NAME="E1:29_1"/>
( not <font color="Maroon" title="c5">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp; <font color="Maroon" title="c5">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">X</font> = <font color="Maroon" title="c5">X</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</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" title="c7">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c8">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font>;<br/><a NAME="E1:29_1_1"/>
 <a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c7">a</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 <b>by </b><i><a class="txt" href="waybel24.html#E1:29"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E2:29_1_1"/>
 <a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c7">a</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <b>by </b><i><a class="ref" href="waybel17.html#T3" title="WAYBEL17:th.3">WAYBEL17:3</a></i>;<br/><a NAME="E3:29_1_1"/>
 <a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c8">b</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 <b>by </b><i><a class="txt" href="waybel24.html#E1:29"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/><b>hence </b><a NAME="E4:29_1_1"/>
 <a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c8">b</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 <b>by </b><i><a class="ref" href="waybel17.html#T3" title="WAYBEL17:th.3">WAYBEL17:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:29_1"/><b>then </b><i><font color="Green" title="E13">A2</font></i>: 
<font color="Maroon" title="c4">f</font> is <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a>
 
<b>by </b><i><a class="ref" href="waybel24.html#T12" target="_self" title="WAYBEL24:th.12">Th12</a></i>;<br/>
<a NAME="E5:29_1"/>
<font color="Maroon" title="c4">f</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c6">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" title="E14">A3</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c6">X</font>,<span class="p1"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span>
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a><br/>

<a NAME="E2:29_1_2"/><i><font color="Green" title="E15">A4</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c3">T</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>
<a NAME="E3:29_1_2"/><i><font color="Green" title="E16">A5</font></i>: 
(  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c6">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> &amp; not  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c6">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T21" title="YELLOW_3:th.21">YELLOW_3:21</a>, <a class="ref" href="yellow_3.html#T22" title="YELLOW_3:th.22">YELLOW_3:22</a></i>;<br/>
<a NAME="E4:29_1_2"/><i><font color="Green" title="E17">A6</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="txt" href="waybel24.html#E1:29_1_2"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="yellow_3.html#T41" title="YELLOW_3:th.41">YELLOW_3:41</a></i>;<br/>
<a NAME="E5:29_1_2"/><i><font color="Green" title="E18">A7</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c2">S</font>
 
<b>by </b><i><a class="txt" href="waybel24.html#E1:29_1_2"><i><font color="Green" title="E14">A3</font></i></a>, <a class="ref" href="yellow_3.html#T41" title="YELLOW_3:th.41">YELLOW_3:41</a></i>;<br/>
<a NAME="E6:29_1_2"/>
 <a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font></span>)</span>,<font color="Maroon" title="c2">S</font></span>)</span> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel24.html#E1:29"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>
<a NAME="E7:29_1_2"/><b>then </b><i><font color="Green" title="E19">A8</font></i>: 
 <a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font></span>)</span>,<font color="Maroon" title="c2">S</font></span>)</span> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a>  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c6">X</font>
 
<b>by </b><i><a class="txt" href="waybel24.html#E3:29_1_2"><i><font color="Green" title="E16">A5</font></i></a>, <a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></i>;<br/>
<i><font color="Green" title="E20">A9</font></i>: <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X</font></span>)</span> = 
<font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="yellow_3.html#T46" title="YELLOW_3:th.46">YELLOW_3:46</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="waybel24.html#T8" target="_self" title="WAYBEL24:th.8">Th8</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<span class="p3">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p4">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel24.html#E4:29_1_2"><i><font color="Green" title="E17">A6</font></i></a>, <a class="txt" href="waybel24.html#E7:29_1_2"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></i>
;<br/>
<a NAME="E9:29_1_2"/>
for <font color="Olive" title="b1">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">T</font>  st <font color="Olive" title="b1">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c6">X</font></span>)</span> holds <br/><font color="Olive" title="b1">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">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" title="c7">b</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">T</font>;<br/>

<b>assume </b><a NAME="E1:29_1_2_2"/>
<font color="Maroon" title="c7">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c6">X</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c8">b1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_1_2_2"/><i><font color="Green" title="E21">A10</font></i>: 
( <font color="Maroon" title="c8">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon" title="c8">b1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c7">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">b1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c9">b1</font> = <font color="Maroon" title="c8">b1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="waybel24.html#E3:29_1_2_2"><i><font color="Green" title="E21">A10</font></i></a></i>;<br/>
<a NAME="E5:29_1_2_2"/><i><font color="Green" title="E22">A11</font></i>: 
( not  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> &amp;  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font> is <a href="waybel_0.html#V1" title="WAYBEL_0:attr.1">directed</a> )
 
<b>by </b><i><a class="ref" href="yellow_3.html#T21" title="YELLOW_3:th.21">YELLOW_3:21</a>, <a class="ref" href="yellow_3.html#T22" title="YELLOW_3:th.22">YELLOW_3:22</a></i>;<br/>
<a NAME="E6:29_1_2_2"/>
 <a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c9">b1</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 
<b>by </b><i><a class="txt" href="waybel24.html#E1:29"><i><font color="Green" title="E12">A1</font></i></a></i>;<br/>
<a NAME="E7:29_1_2_2"/><b>then </b><i><font color="Green" title="E23">A12</font></i>: 
 <a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c9">b1</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font>
 
<b>by </b><i><a class="txt" href="waybel24.html#E5:29_1_2_2"><i><font color="Green" title="E22">A11</font></i></a>, <a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></i>;<br/>
<i><font color="Green" title="E24">A13</font></i>: <font color="Maroon" title="c7">b</font> = 
<span class="p1">(<span class="default"><a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c9">b1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel24.html#E3:29_1_2_2"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="waybel24.html#T9" target="_self" title="WAYBEL24:th.9">Th9</a></i>
<br/>.= 
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c9">b1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p2">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="waybel24.html#E5:29_1_2"><i><font color="Green" title="E18">A7</font></i></a>, <a class="txt" href="waybel24.html#E7:29_1_2_2"><i><font color="Green" title="E23">A12</font></i></a>, <a class="ref" href="waybel_0.html#D31" title="WAYBEL_0:def.31">WAYBEL_0:def 31</a></i>
;<br/>
<a NAME="E9:29_1_2_2"/>
<font color="Maroon" title="c7">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">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" title="E25">A14</font></i>: 
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c9">b1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font></span>)</span>,<font color="Maroon" title="c3">T</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">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" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c9">b1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">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" title="c10">k</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c3">T</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a><br/>

<b>assume </b><a NAME="E1:29_1_2_2_2_1"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c9">b1</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">k1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:29_1_2_2_2_1"/><i><font color="Green" title="E26">A15</font></i>: 
( <font color="Maroon" title="c11">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c9">b1</font></span>)</span> &amp; <font color="Maroon" title="c11">k1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font> &amp; <font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c9">b1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">k1</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">k1</font> = <font color="Maroon" title="c11">k1</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="txt" href="waybel24.html#E3:29_1_2_2_2_1"><i><font color="Green" title="E26">A15</font></i></a></i>;<br/>
<a NAME="E5:29_1_2_2_2_1"/>
<font color="Maroon" title="c10">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> <font color="Maroon" title="c9">b1</font>,<font color="Maroon" title="c12">k1</font>
 
<b>by </b><i><a class="txt" href="waybel24.html#E3:29_1_2_2_2_1"><i><font color="Green" title="E26">A15</font></i></a>, <a class="ref" href="waybel24.html#T7" target="_self" title="WAYBEL24:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E6:29_1_2_2_2_1"/>
<font color="Maroon" title="c10">k</font> <a href="orders_2.html#R1" title="ORDERS_2:pred.1">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel24.html#E4:29_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="txt" href="waybel24.html#E2:29_1_2"><i><font color="Green" title="E15">A4</font></i></a>, <a class="txt" href="waybel24.html#E3:29_1_2_2"><i><font color="Green" title="E21">A10</font></i></a>, <a class="txt" href="waybel24.html#E3:29_1_2_2_2_1"><i><font color="Green" title="E26">A15</font></i></a>, <a class="ref" href="waybel24.html#T17" target="_self" title="WAYBEL24:th.17">Th17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:29_1_2_2_2"/>
<font color="Maroon" title="c7">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">X</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel24.html#E8:29_1_2_2"><i><font color="Green" title="E24">A13</font></i></a>, <a class="txt" href="waybel24.html#E1:29_1_2_2_2"><i><font color="Green" title="E25">A14</font></i></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">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" title="c7">b</font> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">X</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E10:29_1_2"/><b>then </b><i><font color="Green" title="E21">A16</font></i>: 
<span class="p1">(<span class="default"><a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="lattice3.html#R2" title="LATTICE3:pred.2">is_&lt;=_than</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">X</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lattice3.html#D9" title="LATTICE3:def.9">LATTICE3:def 9</a></i>;<br/>
<a NAME="E11:29_1_2"/>
 <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <span class="p1">(<span class="default"><a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c4">f</font>,<span class="p2">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p3">(<span class="default"><a href="yellow_3.html#K5" title="YELLOW_3:func.5">proj2</a> <font color="Maroon" title="c6">X</font></span>)</span></span>)</span></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><a href="yellow_3.html#K4" title="YELLOW_3:func.4">proj1</a> <font color="Maroon" title="c6">X</font></span>)</span>,<font color="Maroon" title="c3">T</font>
 
<b>by </b><i><a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a></i>;<br/>
<a NAME="E12:29_1_2"/><b>then </b><i><font color="Green" title="E22">A17</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a>  <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel24.html#E8:29_1_2"><i><font color="Green" title="E20">A9</font></i></a>, <a class="txt" href="waybel24.html#E10:29_1_2"><i><font color="Green" title="E21">A16</font></i></a>, <a class="ref" href="yellow_0.html#D9" title="YELLOW_0:def.9">YELLOW_0:def 9</a></i>;<br/>
<a NAME="E13:29_1_2"/>
 <a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">X</font></span>)</span> <a href="orders_2.html#R3" title="ORDERS_2:pred.3">&lt;=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#NK3" title="YELLOW_0:NK.3">sup</a> <font color="Maroon" title="c6">X</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel24.html#E4:29_1"><i><font color="Green" title="E13">A2</font></i></a>, <a class="ref" href="waybel17.html#T16" title="WAYBEL17:th.16">WAYBEL17:16</a></i>;<br/>
<b>hence </b><a NAME="E14:29_1_2"/>
(  <a href="yellow_0.html#R1" title="YELLOW_0:pred.1">ex_sup_of</a> <font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">X</font>,<font color="Maroon" title="c3">T</font> &amp;  <a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c6">X</font></span>)</span>,<font color="Maroon" title="c3">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="yellow_0.html#K1" title="YELLOW_0:func.1">"\/"</a> <font color="Maroon" title="c6">X</font>,<span class="p2"><a href="yellow_3.html#K3" title="YELLOW_3:func.3">[:</a><span class="default"><font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c2">S</font></span><a href="yellow_3.html#K3" title="YELLOW_3:func.3">:]</a></span></span>)</span> )
 <b>by </b><i><a class="txt" href="waybel24.html#E12:29_1_2"><i><font color="Green" title="E22">A17</font></i></a>, <a class="ref" href="yellow_0.html#T17" title="YELLOW_0:th.17">YELLOW_0:17</a>, <a class="ref" href="yellow_0.html#D3" title="YELLOW_0:def.3">YELLOW_0:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:29_1"/>
<font color="Maroon" title="c4">f</font> <a href="waybel_0.html#R4" title="WAYBEL_0:pred.4">preserves_sup_of</a> <font color="Maroon" title="c5">X</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:29"/>
<font color="Maroon" title="c4">f</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D37" title="WAYBEL_0:def.37">WAYBEL_0:def 37</a></i>;<br/>


</div>
