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<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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</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="c1">R</font>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font></span>)</span> holds   <a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Olive" title="b1">f</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"><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></span><br/><b>let </b><font color="Maroon" title="c4">f</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">R</font>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c4">f</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"><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></span><br/>



<a NAME="E1:64"/><i><font color="Green" title="E47">A1</font></i>: 
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="c2">S</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font>
 
<b>by </b><i><a class="ref" href="waybel27.html#T22" target="_self" title="WAYBEL27:th.22">Th22</a></i>;<br/>
<a NAME="E2:64"/><b>then </b>
( <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,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="c2">S</font>,<font color="Maroon" title="c3">T</font></span>)</span> &amp;  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,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="c2">S</font>,<font color="Maroon" title="c3">T</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,<span class="p1">(<span class="default"><a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a>, <a class="ref" href="funct_5.html#T63" title="FUNCT_5:th.63">FUNCT_5:63</a></i>;<br/>
<a NAME="E3:64"/><b>then </b>
 <a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font>
 
<b>by </b><i><a class="ref" href="funct_6.html#T20" title="FUNCT_6:th.20">FUNCT_6:20</a></i>;<br/>
<a NAME="E4:64"/><b>then </b>
 <a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font>
 
<b>by </b><i><a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c5">g</font> =  <a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c4">f</font> as   <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> <b>by </b><i><a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c6">a</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> 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="c2">S</font> holds <br/> (  <a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">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="c5">g</font>,<font color="Olive" title="b1">b</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> )</span><br/><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="c2">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">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="c5">g</font>,<font color="Maroon" title="c7">b</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a> )</span><br/><a NAME="E1:64_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K1" title="FUNCT_2:func.1">Funcs</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="txt" href="waybel27.html#E1:64"><i><font color="Green" title="E47">A1</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><a NAME="E2:64_1"/><b>then </b>
 <a href="funct_5.html#K1" title="FUNCT_5:func.1">curry</a> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="ref" href="funct_5.html#T55" title="FUNCT_5:th.55">FUNCT_5:55</a></i>;<br/><a NAME="E3:64_1"/><b>then </b>
 <a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">a</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> <font color="Maroon" title="c6">a</font>
 <b>by </b><i><a class="ref" href="waybel24.html#D1" title="WAYBEL24:def.1">WAYBEL24:def 1</a></i>;<br/><b>hence </b><a NAME="E4:64_1"/>
 <a href="waybel24.html#K1" title="WAYBEL24:func.1">Proj</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c6">a</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="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c7">b</font> is <a href="waybel_0.html#V22" title="WAYBEL_0:attr.22">directed-sups-preserving</a></span><br/><b>reconsider </b><font color="Maroon" title="c8">ST</font> =  <a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c2">S</font>,<font color="Maroon" title="c3">T</font> as  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a> <a href="yellow_0.html#V8" title="YELLOW_0:attr.8">sups-inheriting</a>  <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</a> of <font color="Maroon" title="c3">T</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font> <b>by </b><i><a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a>, <a class="ref" href="waybel27.html#T26" target="_self" title="WAYBEL27:th.26">Th26</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">f'</font> = <font color="Maroon" title="c4">f</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">R</font>,<font color="Maroon" title="c8">ST</font> ;<br/><a NAME="E7:64_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">ST</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font></span>)</span>
 <b>by </b><i><a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></i>;<br/><a NAME="E8:64_1"/><b>then </b><i><font color="Green" title="E48">A2</font></i>: 
 <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c8">ST</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c8">ST</font>
 <b>by </b><i><a class="ref" href="yellow_9.html#D1" title="YELLOW_9:def.1">YELLOW_9:def 1</a></i>;<br/><a NAME="E9:64_1"/>
 <a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c8">ST</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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="ref" href="waybel21.html#T10" title="WAYBEL21:th.10">WAYBEL21:10</a></i>;<br/><a NAME="E10:64_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="yellow_9.html#K1" title="YELLOW_9:func.1">incl</a> <font color="Maroon" title="c8">ST</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">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="waybel20.html#T29" title="WAYBEL20:th.29">WAYBEL20:29</a></i>;<br/><a NAME="E11:64_1"/><b>then </b>
<font color="Maroon" title="c4">f</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="c1">R</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">T</font> <a href="yellow_1.html#K6" title="YELLOW_1:func.6">|^</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S</font></span>)</span>
 <b>by </b><i><a class="txt" href="waybel27.html#E8:64_1"><i><font color="Green" title="E48">A2</font></i></a>, <a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>;<br/><a NAME="E12:64_1"/><b>then </b><i><font color="Green" title="E49">A3</font></i>: 
<span class="p1">(<span class="default"><a href="funct_6.html#K10" title="FUNCT_6:func.10">commute</a> <font color="Maroon" title="c4">f</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</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="c1">R</font>,<font color="Maroon" title="c3">T</font>
 <b>by </b><i><a class="ref" href="waybel27.html#T38" target="_self" title="WAYBEL27:th.38">Th38</a></i>;<br/><a NAME="E13:64_1"/>
 <a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c5">g</font>,<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="funct_5.html#K3" title="FUNCT_5:func.3">curry'</a> <font color="Maroon" title="c5">g</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">b</font>
 <b>by </b><i><a class="ref" href="waybel24.html#D2" title="WAYBEL24:def.2">WAYBEL24:def 2</a></i>;<br/><b>hence </b><a NAME="E14:64_1"/>
 <a href="waybel24.html#K2" title="WAYBEL24:func.2">Proj</a> <font color="Maroon" title="c5">g</font>,<font color="Maroon" title="c7">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="waybel27.html#E12:64_1"><i><font color="Green" title="E49">A3</font></i></a>, <a class="ref" href="funct_6.html#D12" title="FUNCT_6:def.12">FUNCT_6:def 12</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:64"/>
 <a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c4">f</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"><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>
 <b>by </b><i><a class="ref" href="waybel24.html#T18" title="WAYBEL24:th.18">WAYBEL24:18</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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