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<b>let </b><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font>, <font color="Maroon" title="c3">T1</font>, <font color="Maroon" title="c4">T2</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="c5">f</font> be   <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="c6">g</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">T1</font>,<font color="Maroon" title="c4">T2</font>;<br/>





<b>assume </b><a NAME="E1:53"/><i><font color="Green" title="E38">A1</font></i>: 
( <font color="Maroon" title="c5">f</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> &amp; <font color="Maroon" title="c6">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> )
 ;<br/>

<a NAME="E2:53"/><b>then </b><i><font color="Green" title="E39">A2</font></i>: 
( <font color="Maroon" title="c5">f</font> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">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> &amp; <font color="Maroon" title="c6">g</font> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c3">T1</font>,<font color="Maroon" title="c4">T2</font> )
 
<b>by </b><i><a class="ref" href="waybel13.html#T20" title="WAYBEL13:th.20">WAYBEL13:20</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">f'</font> being  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</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="c1">S1</font><b> such that </b><br/><a NAME="E4:53"/><i><font color="Green" title="E40">A3</font></i>: 
( <font color="Maroon" title="c5">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">f'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c2">S2</font> &amp; <font color="Maroon" title="c7">f'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">f</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">S1</font> )
 <b>by </b><i><a class="txt" href="waybel27.html#E1:53"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="yellow16.html#T17" title="YELLOW16:th.17">YELLOW16:17</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">g'</font> being  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">T2</font>,<font color="Maroon" title="c3">T1</font><b> such that </b><br/><a NAME="E6:53"/><i><font color="Green" title="E41">A4</font></i>: 
( <font color="Maroon" title="c6">g</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c8">g'</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c4">T2</font> &amp; <font color="Maroon" title="c8">g'</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">g</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c3">T1</font> )
 <b>by </b><i><a class="txt" href="waybel27.html#E1:53"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="yellow16.html#T17" title="YELLOW16:th.17">YELLOW16:17</a></i>;<br/>
<a NAME="E7:53"/><i><font color="Green" title="E42">A5</font></i>: 
 <a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</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 <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c3">T1</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c4">T2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="waybel27.html#E2:53"><i><font color="Green" title="E39">A2</font></i></a>, <a class="ref" href="waybel27.html#T30" target="_self" title="WAYBEL27:th.30">Th30</a></i>;<br/>
<a NAME="E8:53"/>
( <font color="Maroon" title="c7">f'</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> &amp; <font color="Maroon" title="c8">g'</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> )
 
<b>by </b><i><a class="txt" href="waybel27.html#E2:53"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="waybel27.html#E4:53"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="waybel27.html#E6:53"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="yellow16.html#T17" title="YELLOW16:th.17">YELLOW16:17</a></i>;<br/>
<a NAME="E9:53"/><b>then </b><i><font color="Green" title="E43">A6</font></i>: 
( <font color="Maroon" title="c7">f'</font> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">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="c1">S1</font> &amp; <font color="Maroon" title="c8">g'</font> is  <a href="waybel_0.html#V18" title="WAYBEL_0:attr.18">sups-preserving</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <font color="Maroon" title="c4">T2</font>,<font color="Maroon" title="c3">T1</font> )
 
<b>by </b><i><a class="ref" href="waybel13.html#T20" title="WAYBEL13:th.20">WAYBEL13:20</a></i>;<br/>
<a NAME="E10:53"/><b>then </b><i><font color="Green" title="E44">A7</font></i>: 
 <a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c7">f'</font>,<font color="Maroon" title="c8">g'</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 <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c4">T2</font></span>)</span>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c3">T1</font></span>)</span>
 
<b>by </b><i><a class="ref" href="waybel27.html#T30" target="_self" title="WAYBEL27:th.30">Th30</a></i>;<br/>
<i><font color="Green" title="E45">A8</font></i>: <span class="p1">(<span class="default"><a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</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">f'</font>,<font color="Maroon" title="c8">g'</font></span>)</span> = 
 <a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c1">S1</font></span>)</span>,<span class="p1">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c4">T2</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel27.html#E2:53"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="waybel27.html#E4:53"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="waybel27.html#E6:53"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="waybel27.html#E9:53"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="waybel27.html#T28" target="_self" title="WAYBEL27:th.28">Th28</a></i>
<br/>.= 
 <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">S1</font>,<font color="Maroon" title="c4">T2</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel27.html#T29" target="_self" title="WAYBEL27:th.29">Th29</a></i>
;<br/>
<span class="p1">(<span class="default"><a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c7">f'</font>,<font color="Maroon" title="c8">g'</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="c5">f</font>,<font color="Maroon" title="c6">g</font></span>)</span> = 
 <a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <span class="p1">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c2">S2</font></span>)</span>,<span class="p1">(<span class="default"><a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <font color="Maroon" title="c3">T1</font></span>)</span>
<b>by </b><i><a class="txt" href="waybel27.html#E2:53"><i><font color="Green" title="E39">A2</font></i></a>, <a class="txt" href="waybel27.html#E4:53"><i><font color="Green" title="E40">A3</font></i></a>, <a class="txt" href="waybel27.html#E6:53"><i><font color="Green" title="E41">A4</font></i></a>, <a class="txt" href="waybel27.html#E9:53"><i><font color="Green" title="E43">A6</font></i></a>, <a class="ref" href="waybel27.html#T28" target="_self" title="WAYBEL27:th.28">Th28</a></i>
<br/>.= 
 <a href="grcat_1.html#K7" title="GRCAT_1:func.7">id</a> <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c2">S2</font>,<font color="Maroon" title="c3">T1</font></span>)</span>
<b>by </b><i><a class="ref" href="waybel27.html#T29" target="_self" title="WAYBEL27:th.29">Th29</a></i>
;<br/>
<b>hence </b><a NAME="E13:53"/>
 <a href="waybel27.html#K4" title="WAYBEL27:func.4">UPS</a> <font color="Maroon" title="c5">f</font>,<font color="Maroon" title="c6">g</font> is <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>
 <b>by </b><i><a class="txt" href="waybel27.html#E7:53"><i><font color="Green" title="E42">A5</font></i></a>, <a class="txt" href="waybel27.html#E10:53"><i><font color="Green" title="E44">A7</font></i></a>, <a class="txt" href="waybel27.html#E11:53"><i><font color="Green" title="E45">A8</font></i></a>, <a class="ref" href="yellow16.html#T17" title="YELLOW16:th.17">YELLOW16:17</a></i>;<br/>


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