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

<b>let </b><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font>, <font color="Maroon" title="c3">S3</font>, <font color="Maroon" title="c4">T1</font>, <font color="Maroon" title="c5">T2</font>, <font color="Maroon" title="c6">T3</font> be  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="orders_2.html#NM1" title="ORDERS_2:NM.1">Poset</a>;<br/><b>let </b><font color="Maroon" title="c7">f1</font> be  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c3">S3</font>;<br/><b>let </b><font color="Maroon" title="c8">f2</font> be  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c2">S2</font>;<br/><b>let </b><font color="Maroon" title="c9">g1</font> be  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">T1</font>,<font color="Maroon" title="c5">T2</font>;<br/><b>let </b><font color="Maroon" title="c10">g2</font> be  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c5">T2</font>,<font color="Maroon" title="c6">T3</font>;<br/>









<b>reconsider </b><font color="Maroon" title="c11">F</font> = <font color="Maroon" title="c7">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f2</font> as  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c3">S3</font> <b>by </b><i><a class="ref" href="waybel20.html#T29" title="WAYBEL20:th.29">WAYBEL20:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">G</font> = <font color="Maroon" title="c10">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">g1</font> as  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">T1</font>,<font color="Maroon" title="c6">T3</font> <b>by </b><i><a class="ref" href="waybel20.html#T29" title="WAYBEL20:th.29">WAYBEL20:29</a></i>;<br/>
<a NAME="E3:46"/>
for <font color="Olive" title="b1">h</font> being <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">S3</font>,<font color="Maroon" title="c4">T1</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c8">f2</font>,<font color="Maroon" title="c10">g2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c9">g1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Olive" title="b1">h</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">F</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="46_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">h</font> be  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">S3</font>,<font color="Maroon" title="c4">T1</font>;<br/>

<a NAME="E1:46_1"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c9">g1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c3">S3</font>,<font color="Maroon" title="c4">T1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:46_1"/><b>then </b>
<font color="Maroon" title="c13">h</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="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c9">g1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></i>;<br/>
<b>then </b><i><font color="Green" title="E32">A1</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c8">f2</font>,<font color="Maroon" title="c10">g2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c9">g1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">h</font> = 
<span class="p1">(<span class="default"><a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c8">f2</font>,<font color="Maroon" title="c10">g2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c9">g1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">h</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c8">f2</font>,<font color="Maroon" title="c10">g2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">h</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">f1</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel27.html#D5" target="_self" title="WAYBEL27:def.5">Def5</a></i>
;<br/>
<a NAME="E4:46_1"/>
<font color="Maroon" title="c9">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">h</font> is  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">S3</font>,<font color="Maroon" title="c5">T2</font>
 
<b>by </b><i><a class="ref" href="waybel20.html#T29" title="WAYBEL20:th.29">WAYBEL20:29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c14">ghf</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">h</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">f1</font> as  <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c5">T2</font> <b>by </b><i><a class="ref" href="waybel20.html#T29" title="WAYBEL20:th.29">WAYBEL20:29</a></i>;<br/>
<b>thus </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c8">f2</font>,<font color="Maroon" title="c10">g2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c9">g1</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">ghf</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f2</font>
<b>by </b><i><a class="txt" href="waybel27.html#E3:46_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="ref" href="waybel27.html#D5" target="_self" title="WAYBEL27:def.5">Def5</a></i>
<br/>.= 
<font color="Maroon" title="c10">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">h</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon" title="c10">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c13">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">f1</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon" title="c10">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c13">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">f1</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon" title="c10">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">h</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon" title="c10">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">h</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">g1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">h</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">G</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c13">h</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">F</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:46"/>
<span class="p1">(<span class="default"><a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c8">f2</font>,<font color="Maroon" title="c10">g2</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c7">f1</font>,<font color="Maroon" title="c9">g1</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f1</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">f2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c10">g2</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">g1</font></span>)</span>
 <b>by </b><i><a class="ref" href="waybel27.html#D5" target="_self" title="WAYBEL27:def.5">Def5</a></i>;<br/>


</div>
