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<b>let </b><font color="Maroon">S1</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">S2</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">T1</font> be  <a href="lattice3.html#V3">complete</a> <a href="waybel_0.html#NM8">LATTICE</a>, <font color="Maroon">T2</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 <font color="Maroon">S1</font>,<font color="Maroon">S2</font>;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T1</font>,<font color="Maroon">T2</font>;<br/>





<b>assume </b><a NAME="E1:53"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">f</font> is <a href="waybel_0.html#V23">isomorphic</a> &amp; <font color="Maroon">g</font> is <a href="waybel_0.html#V23">isomorphic</a> )
 ;<br/>

<a NAME="E2:53"/><b>then </b><i><font color="Green">E38</font></i>: 
( <font color="Maroon">f</font> is  <a href="waybel_0.html#V18">sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S1</font>,<font color="Maroon">S2</font> &amp; <font color="Maroon">g</font> is  <a href="waybel_0.html#V18">sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T1</font>,<font color="Maroon">T2</font> )
 
<b>by </b><i><a class="ref" href="waybel13.html#T20">WAYBEL13:20</a></i>;<br/>
<b>consider </b><font color="Maroon">f'</font> being  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S2</font>,<font color="Maroon">S1</font><b> such that </b><br/><a NAME="E4:53"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">f</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f'</font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">S2</font> &amp; <font color="Maroon">f'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">S1</font> )
 <b>by </b><i><a class="txt" href="waybel27.html#E2:53"><i><font color="Green">E38</font></i></a>, <a class="ref" href="yellow16.html#T17">YELLOW16:17</a></i>;<br/>
<b>consider </b><font color="Maroon">g'</font> being  <a href="orders_3.html#V5">monotone</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T2</font>,<font color="Maroon">T1</font><b> such that </b><br/><a NAME="E6:53"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g'</font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">T2</font> &amp; <font color="Maroon">g'</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font> <a href="funct_2.html#R2">=</a>  <a href="grcat_1.html#K7">id</a> <font color="Maroon">T1</font> )
 <b>by </b><i><a class="txt" href="waybel27.html#E2:53"><i><font color="Green">E38</font></i></a>, <a class="ref" href="yellow16.html#T17">YELLOW16:17</a></i>;<br/>
<a NAME="E7:53"/><i><font color="Green">E51</font></i>: 
 <a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> is  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S2</font>,<font color="Maroon">T1</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S1</font>,<font color="Maroon">T2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel27.html#E4:53"><i><font color="Green">E39</font></i></a>, </i>;<br/>
<a NAME="E8:53"/>
( <font color="Maroon">f'</font> is <a href="waybel_0.html#V23">isomorphic</a> &amp; <font color="Maroon">g'</font> is <a href="waybel_0.html#V23">isomorphic</a> )
 
<b>by </b><i><a class="txt" href="waybel27.html#E4:53"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel27.html#T1" target="_self">Th1</a>, , <a class="ref" href="yellow16.html#T17">YELLOW16:17</a></i>;<br/>
<a NAME="E9:53"/><b>then </b><i><font color="Green">E54</font></i>: 
( <font color="Maroon">f'</font> is  <a href="waybel_0.html#V18">sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">S2</font>,<font color="Maroon">S1</font> &amp; <font color="Maroon">g'</font> is  <a href="waybel_0.html#V18">sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">T2</font>,<font color="Maroon">T1</font> )
 
<b>by </b><i><a class="ref" href="waybel13.html#T20">WAYBEL13:20</a></i>;<br/>
<a NAME="E10:53"/><b>then </b><i><font color="Green">E55</font></i>: 
 <a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">f'</font>,<font color="Maroon">g'</font> is  <a href="waybel_0.html#V22">directed-sups-preserving</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S1</font>,<font color="Maroon">T2</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S2</font>,<font color="Maroon">T1</font></span>)</span>
 
<b>by </b><i/>;<br/>
<i><font color="Green">E56</font></i>: <span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">f</font>,<font color="Maroon">g</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">f'</font>,<font color="Maroon">g'</font></span>)</span> = 
 <a href="waybel27.html#K4" target="_self">UPS</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S1</font></span>)</span>,<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">T2</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel27.html#E4:53"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel27.html#T1" target="_self">Th1</a>, , , <a class="ref" href="waybel27.html#D1" target="_self">Def1</a></i>
<br/>.= 
 <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S1</font>,<font color="Maroon">T2</font></span>)</span>
<b>by </b><i/>
;<br/>
<span class="p1">(<span class="default"><a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">f'</font>,<font color="Maroon">g'</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">f</font>,<font color="Maroon">g</font></span>)</span> = 
 <a href="waybel27.html#K4" target="_self">UPS</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">S2</font></span>)</span>,<span class="p1">(<span class="default"><a href="grcat_1.html#K7">id</a> <font color="Maroon">T1</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel27.html#E4:53"><i><font color="Green">E39</font></i></a>, <a class="ref" href="waybel27.html#T1" target="_self">Th1</a>, , , <a class="ref" href="waybel27.html#D1" target="_self">Def1</a></i>
<br/>.= 
 <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="waybel27.html#K2" target="_self">UPS</a> <font color="Maroon">S2</font>,<font color="Maroon">T1</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E13:53"/>
 <a href="waybel27.html#K4" target="_self">UPS</a> <font color="Maroon">f</font>,<font color="Maroon">g</font> is <a href="waybel_0.html#V23">isomorphic</a>
 <b>by </b><i>, , , <a class="ref" href="yellow16.html#T17">YELLOW16:17</a></i>;<br/>


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