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<div class="add">

<b>let </b><font color="Maroon">S1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>, <font color="Maroon">S2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>, <font color="Maroon">S3</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>, <font color="Maroon">T1</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>, <font color="Maroon">T2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>, <font color="Maroon">T3</font> be  non <a href="struct_0.html#V3">empty</a> <a href="orders_2.html#NM1">Poset</a>;<br/><b>let </b><font color="Maroon">f1</font> be  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S2</font>,<font color="Maroon">S3</font>;<br/><b>let </b><font color="Maroon">f2</font> be  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S1</font>,<font color="Maroon">S2</font>;<br/><b>let </b><font color="Maroon">g1</font> be  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T1</font>,<font color="Maroon">T2</font>;<br/><b>let </b><font color="Maroon">g2</font> be  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T2</font>,<font color="Maroon">T3</font>;<br/>









<b>reconsider </b><font color="Maroon">F</font> = <font color="Maroon">f1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f2</font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S1</font>,<font color="Maroon">S3</font> <b>by </b><i><a class="ref" href="waybel20.html#T29">WAYBEL20:29</a></i>;<br/>
<b>reconsider </b><font color="Maroon">G</font> = <font color="Maroon">g2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g1</font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T1</font>,<font color="Maroon">T3</font> <b>by </b><i><a class="ref" href="waybel20.html#T29">WAYBEL20:29</a></i>;<br/>
<a NAME="E3:46"/>
for <font color="Olive">h</font> being <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S3</font>,<font color="Maroon">T1</font> holds  <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">f2</font>,<font color="Maroon">g2</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">f1</font>,<font color="Maroon">g1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="partfun1.html#K1">*</a> <font color="Olive">h</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">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">h</font> be  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S3</font>,<font color="Maroon">T1</font>;<br/>

<a NAME="E1:46_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">f1</font>,<font color="Maroon">g1</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">S3</font>,<font color="Maroon">T1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:46_1"/><b>then </b>
<font color="Maroon">h</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> <font color="Maroon">f1</font>,<font color="Maroon">g1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>then </b><i><font color="Green">E36</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">f2</font>,<font color="Maroon">g2</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">f1</font>,<font color="Maroon">g1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">f2</font>,<font color="Maroon">g2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">f1</font>,<font color="Maroon">g1</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">f2</font>,<font color="Maroon">g2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span>
<b>by </b><i/>
;<br/>
<a NAME="E4:46_1"/>
<font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font> is  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S3</font>,<font color="Maroon">T2</font>
 
<b>by </b><i><a class="ref" href="waybel20.html#T29">WAYBEL20:29</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">ghf</font> = <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> as  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S2</font>,<font color="Maroon">T2</font> <b>by </b><i><a class="ref" href="waybel20.html#T29">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" target="_self">UPS</a> <font color="Maroon">f2</font>,<font color="Maroon">g2</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">f1</font>,<font color="Maroon">g1</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">h</font> = 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">ghf</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f2</font>
<b>by </b><i>, </i>
<br/>.= 
<font color="Maroon">g2</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">g2</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">g2</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">g2</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><font color="Maroon">f1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f2</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<font color="Maroon">g2</font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">f1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">g1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f2</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">h</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">F</font>
<b>by </b><i><a class="ref" href="relat_1.html#T55">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" target="_self">UPS</a> <font color="Maroon">f2</font>,<font color="Maroon">g2</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">f1</font>,<font color="Maroon">g1</font></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="waybel27.html#K4" target="_self">UPS</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">g2</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g1</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
