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<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> $1;<br/>
<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being  <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">R</font>,<span class="p2">(<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></span>)</span>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <span class="p2"><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>)</span> st <br/>( <font color="Olive" title="b1">f</font> is  <a href="waybel27.html#V1" title="WAYBEL27:attr.1">uncurrying</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>  )</span><br/>

<span class="kw">consider </span><font color="Maroon" title="c4">F</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of 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="c1">R</font>,<span class="p2">(<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></span>)</span><span class="kw"> such that </span><br/><a NAME="E2:65"/><span class="lab"><font color="Green" title="E48">A1</font></span>: 
for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">R</font>,<span class="p2">(<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></span>)</span> holds  <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">f</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></span>
<a NAME="E3:65"/><span class="lab"><font color="Green" title="E49">A2</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font> <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="c1">R</font>,<span class="p2">(<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></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E4:65"/><span class="lab"><font color="Green" title="E50">A3</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</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"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <span class="p2"><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>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font> or <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <span class="p2"><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>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:65_1"/>
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c4">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <span class="p2"><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>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:65_1"/><span class="lab"><font color="Green" title="E51">A4</font></span>: 
<font color="Maroon" title="c6">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c4">F</font>
 <span class="kw">and </span><br/><a NAME="E4:65_1"/><span class="lab"><font color="Green" title="E52">A5</font></span>: 
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">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>,<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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E3:65_1"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></span>;<br/>
<a NAME="E6:65_1"/>
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Maroon" title="c7">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E3:65_1"><span class="lab"><font color="Green" title="E51">A4</font></span></a>, <a class="txt" href="#E4:65_1"><span class="lab"><font color="Green" title="E52">A5</font></span></a></span>;<br/>
<a NAME="E7:65_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">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"><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 class="kw">by</span> <span class="lab"><a class="ref" href="waybel27.html#T45" target="_self" title="WAYBEL27:th.45">Th45</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:65_1"/>
<font color="Maroon" title="c5">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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> <span class="p2"><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>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">F</font> = <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">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">R</font>,<span class="p2">(<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></span>)</span>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <span class="p2"><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>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c5">F</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">F</font> is  <a href="waybel27.html#V1" title="WAYBEL27:attr.1">uncurrying</a>  &amp; <font color="Maroon" title="c5">F</font> is  <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E6:65"/><span class="lab"><font color="Green" title="E51">A6</font></span>: 
<font color="Maroon" title="c5">F</font> is  <a href="waybel27.html#V1" title="WAYBEL27:attr.1">uncurrying</a> 
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">F</font> is  <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> </span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel27.html#D1" target="_self" title="WAYBEL27:def.1">WAYBEL27:def 1</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">F</font> holds <br/><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Olive" title="b1">f</font></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">F</font> implies <font color="Maroon" title="c6">x</font> is   <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> )</span><br/><span class="kw">assume </span><a NAME="E1:65_2_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c5">F</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">x</font> is   <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a></span><br/><a NAME="E2:65_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">x</font> is   <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:65_2_1"/>
<font color="Maroon" title="c6">x</font> is   <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a>  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:65_2"/>
for <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c5">F</font> holds <br/><font color="Maroon" title="c5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_5.html#K2" title="FUNCT_5:func.2">uncurry</a> <font color="Olive" title="b1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E48">A1</font></span></a>, <a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E49">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>) <span class="kw">-&gt; </span>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="funct_5.html#K1" title="FUNCT_5:func.1">curry</a> $1;<br/>
<span class="kw">consider </span><font color="Maroon" title="c6">G</font> being   <a href="pboole.html#NM1" title="PBOOLE:NM.1">ManySortedSet</a> of 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> <span class="p2"><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>)</span><span class="kw"> such that </span><br/><a NAME="E8:65"/><span class="lab"><font color="Green" title="E52">A7</font></span>: 
for <font color="Olive" title="b1">f</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <span class="p2"><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>)</span> holds  <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">f</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="pboole.html#S5" title="PBOOLE:sch.5">PBOOLE:sch 5</a>();<br/></span>
<a NAME="E9:65"/><span class="lab"><font color="Green" title="E53">A8</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">G</font> <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> <span class="p2"><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>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="partfun1.html#D4" title="PARTFUN1:def.4">PARTFUN1:def 4</a></span>;<br/>
<a NAME="E10:65"/><span class="lab"><font color="Green" title="E54">A9</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</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"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">R</font>,<span class="p2">(<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></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">g</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font> or <font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">R</font>,<span class="p2">(<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></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:65_3"/>
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c6">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">R</font>,<span class="p2">(<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></span>)</span></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">f</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:65_3"/><span class="lab"><font color="Green" title="E55">A10</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c6">G</font>
 <span class="kw">and </span><br/><a NAME="E4:65_3"/><span class="lab"><font color="Green" title="E56">A11</font></span>: 
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">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 <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 class="kw">by</span> <span class="lab"><a class="txt" href="#E9:65"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E3:65_3"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></span>;<br/>
<a NAME="E6:65_3"/>
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_5.html#K1" title="FUNCT_5:func.1">curry</a> <font color="Maroon" title="c9">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:65"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E9:65"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E3:65_3"><span class="lab"><font color="Green" title="E55">A10</font></span></a>, <a class="txt" href="#E4:65_3"><span class="lab"><font color="Green" title="E56">A11</font></span></a></span>;<br/>
<a NAME="E7:65_3"/><span class="kw">then </span>
<font color="Maroon" title="c7">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 <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel27.html#T46" target="_self" title="WAYBEL27:th.46">Th46</a></span>;<br/>
<span class="kw">hence </span><a NAME="E8:65_3"/>
<font color="Maroon" title="c7">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c1">R</font>,<span class="p2">(<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></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">G</font> = <font color="Maroon" title="c6">G</font> as   <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> <span class="p2"><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>)</span>,<span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">R</font>,<span class="p2">(<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></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:65"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></span>;<br/>
<a NAME="E12:65"/>
 <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> is   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></span>;<br/>
<a NAME="E13:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E55">A12</font></span>: 
<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="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="c1">R</font> is   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</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> <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="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow16.html#T41" title="YELLOW16:th.41">YELLOW16:41</a></span>;<br/>
<a NAME="E14:65"/>
 <a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <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> is   <a href="yellow_0.html#V4" title="YELLOW_0:attr.4">full</a>   <a href="yellow_0.html#M1" title="YELLOW_0:mode.1">SubRelStr</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="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="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></span>;<br/>
<a NAME="E15:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A13</font></span>: 
 <a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <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> is  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#M1" title="YELLOW_0:mode.1">SubRelStr</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> <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="c1">R</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:65"><span class="lab"><font color="Green" title="E55">A12</font></span></a>, <a class="ref" href="yellow16.html#T28" title="YELLOW16:th.28">YELLOW16:28</a></span>;<br/>
<a NAME="E16:65"/>
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> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></span>;<br/>
<a NAME="E17:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A14</font></span>: 
 <a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <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> is  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#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> <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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></span>;<br/>
<a NAME="E18:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A15</font></span>: 
<font color="Maroon" title="c5">F</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:65"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E15:65"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="ref" href="waybel27.html#T20" target="_self" title="WAYBEL27:th.20">Th20</a></span>;<br/>
<a NAME="E19:65"/><span class="lab"><font color="Green" title="E59">A16</font></span>: 
<font color="Maroon" title="c7">G</font> is  <a href="waybel27.html#V2" title="WAYBEL27:attr.2">currying</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_4"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="waybel27.html#D2" target="_self" title="WAYBEL27:def.2">WAYBEL27:def 2</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">G</font> holds <br/><font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_5.html#K1" title="FUNCT_5:func.1">curry</a> <font color="Olive" title="b1">f</font></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c8">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">G</font> implies ( <font color="Maroon" title="c8">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c8">x</font> is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> ) )</span><br/><span class="kw">assume </span><a NAME="E1:65_4_1"/>
<font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c7">G</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c8">x</font> is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> )</span><br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">f</font> = <font color="Maroon" title="c8">x</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="waybel27.html#D4" target="_self" title="WAYBEL27:def.4">Def4</a></span>;<br/> <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c9">f</font> = 
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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>
<br/>.= 
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_3.html#D2" title="YELLOW_3:def.2">YELLOW_3:def 2</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:65_4_1"/>
( <font color="Maroon" title="c8">x</font> is   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> &amp;  <a href="relat_1.html#K1" title="RELAT_1:func.1">proj1</a> <font color="Maroon" title="c8">x</font> is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E2:65_4"/>
for <font color="Olive" title="b1">f</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c7">G</font> holds <br/><font color="Maroon" title="c7">G</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_5.html#K1" title="FUNCT_5:func.1">curry</a> <font color="Olive" title="b1">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:65"><span class="lab"><font color="Green" title="E52">A7</font></span></a>, <a class="txt" href="#E9:65"><span class="lab"><font color="Green" title="E53">A8</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E20:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A17</font></span>: 
<font color="Maroon" title="c7">G</font> is  <a href="orders_3.html#V5" title="ORDERS_3:attr.5">monotone</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:65"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="txt" href="#E17:65"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="ref" href="waybel27.html#T21" target="_self" title="WAYBEL27:th.21">Th21</a></span>;<br/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><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> <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="c1">R</font></span>)</span> = 
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">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"><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 class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T28" title="YELLOW_1:th.28">YELLOW_1:28</a></span>
<br/>.= 
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">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#K9" title="FUNCT_2:func.9">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T28" title="YELLOW_1:th.28">YELLOW_1:28</a></span>
;<br/>
<a NAME="E22:65"/><span class="kw">then </span>
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="c1">R</font>,<span class="p2">(<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></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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#K9" title="FUNCT_2:func.9">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:65"><span class="lab"><font color="Green" title="E56">A13</font></span></a>, <a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></span>;<br/>
<a NAME="E23:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A18</font></span>: 
<font color="Maroon" title="c7">G</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="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <font color="Maroon" title="c1">R</font>,<span class="p2">(<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></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E4:65"><span class="lab"><font color="Green" title="E50">A3</font></span></a>, <a class="txt" href="#E9:65"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E6:65"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E19:65"><span class="lab"><font color="Green" title="E59">A16</font></span></a>, <a class="ref" href="waybel27.html#T4" target="_self" title="WAYBEL27:th.4">Th4</a></span>;<br/>
<a NAME="E24:65"/>
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> <span class="p2"><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></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="yellow_1.html#T28" title="YELLOW_1:th.28">YELLOW_1:28</a></span>;<br/>
<a NAME="E25:65"/><span class="kw">then </span>
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> <span class="p2"><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>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:65"><span class="lab"><font color="Green" title="E57">A14</font></span></a>, <a class="ref" href="yellow_0.html#D13" title="YELLOW_0:def.13">YELLOW_0:def 13</a></span>;<br/>
<a NAME="E26:65"/><span class="kw">then </span>
<font color="Maroon" title="c5">F</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">G</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p1">(<span class="default"><a href="waybel27.html#K2" title="WAYBEL27:func.2">UPS</a> <span class="p2"><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>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65"><span class="lab"><font color="Green" title="E49">A2</font></span></a>, <a class="txt" href="#E9:65"><span class="lab"><font color="Green" title="E53">A8</font></span></a>, <a class="txt" href="#E10:65"><span class="lab"><font color="Green" title="E54">A9</font></span></a>, <a class="txt" href="#E6:65"><span class="lab"><font color="Green" title="E51">A6</font></span></a>, <a class="txt" href="#E19:65"><span class="lab"><font color="Green" title="E59">A16</font></span></a>, <a class="ref" href="waybel27.html#T5" target="_self" title="WAYBEL27:th.5">Th5</a></span>;<br/>
<span class="kw">hence </span><a NAME="E27:65"/>
<font color="Maroon" title="c5">F</font> is  <a href="waybel_0.html#V23" title="WAYBEL_0:attr.23">isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:65"><span class="lab"><font color="Green" title="E61">A18</font></span></a>, <a class="txt" href="#E18:65"><span class="lab"><font color="Green" title="E58">A15</font></span></a>, <a class="txt" href="#E20:65"><span class="lab"><font color="Green" title="E60">A17</font></span></a>, <a class="ref" href="yellow16.html#T17" title="YELLOW16:th.17">YELLOW16:17</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
