<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</font>, <font color="Maroon" title="c3">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> ;<br/><b>let </b><font color="Maroon" title="c4">V</font> be   <a href="hilbert3.html#NM1" title="HILBERT3:NM.1">SetValuation</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="hilbert3.html#D7" target="_self" title="HILBERT3:def.7">HILBERT3:def 7</a><br/>







<a NAME="E1:55"/><i><font color="Green" title="E39">A1</font></i>: 
(  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c2">q</font></span>)</span> &amp;  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c3">r</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  =  <a href="funct_6.html#K7" title="FUNCT_6:func.7">Frege</a> $1;<br/>
<a NAME="E2:55"/><i><font color="Green" title="E40">A2</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span>;<br/>

<a NAME="E1:55_1"/>
<font color="Maroon" title="c5">x</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/>
<a NAME="E2:55_1"/><b>then </b>
<font color="Maroon" title="c5">x</font> is    <a href="fraenkel.html#M2" title="FRAENKEL:mode.2">Element</a> of  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p2">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c2">q</font></span>)</span>,<span class="p2">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c3">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/>
<a NAME="E3:55_1"/><b>then </b>
 <a href="pralg_2.html#K2" title="PRALG_2:func.2">Frege</a> <font color="Maroon" title="c5">x</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span>, <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E1:55"><i><font color="Green" title="E39">A1</font></i></a>, <a class="ref" href="hilbert3.html#T24" target="_self" title="HILBERT3:th.24">Th24</a></i>;<br/>
<a NAME="E4:55_1"/><b>then </b>
 <a href="pralg_2.html#K2" title="PRALG_2:func.2">Frege</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E5:55_1"/>
 <a href="pralg_2.html#K2" title="PRALG_2:func.2">Frege</a> <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span>, <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:55"/><i><font color="Green" title="E41">A3</font></i>: 
for <font color="Olive" title="b1">x</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> holds  <font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8" title="FUNCT_2:sch.8">FUNCT_2:sch 8</a>(<a class="txt" href="hilbert3.html#E2:55"><i><font color="Green" title="E40">A2</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c5">F</font>
;<br/>

<b>let </b><font color="Maroon" title="c6">P</font> be    <a href="hilbert3.html#M1" title="HILBERT3:mode.1">Permutation</a> of <font color="Maroon" title="c4">V</font>;<br/>

<a NAME="E5:55"/>
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<a NAME="E6:55"/><b>then </b>
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/>
<b>hence </b><a NAME="E7:55"/>
<font color="Maroon" title="c5">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> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">f</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span>;<br/><b>reconsider </b><font color="Maroon" title="c8">X</font> = <span class="p1">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span>, <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span> ;<br/><a NAME="E2:55_2"/>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span>
 ;<br/><a NAME="E3:55_2"/><b>then </b>
<font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c9">ff</font> = <font color="Maroon" title="c7">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font>, <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/><a NAME="E5:55_2"/><i><font color="Green" title="E42">A4</font></i>: 
 <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c2">q</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c3">r</font></span>)</span>
 <b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/><b>then </b><i><font color="Green" title="E43">A5</font></i>:  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c9">ff</font></span>)</span> = 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c2">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hilbert3.html#T6" target="_self" title="HILBERT3:th.6">Th6</a></i>
<br/>.= 
 <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c2">q</font></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T20" title="CARD_3:th.20">CARD_3:20</a></i>
;<br/><a NAME="E7:55_2"/><b>then </b><i><font color="Green" title="E44">A6</font></i>: 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c9">ff</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/><b>set </b><font color="Maroon" title="c10">Yf</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font>;<br/><a NAME="E8:55_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">ff</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="hilbert3.html#T38" target="_self" title="HILBERT3:th.38">Th38</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c11">h</font> =  <a href="pralg_2.html#K2" title="PRALG_2:func.2">Frege</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font></span>)</span> as  <a href="funcop_1.html#V1" title="FUNCOP_1:attr.1">Function-yielding</a> <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span>, <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span> <b>by </b><i><a class="txt" href="hilbert3.html#E1:55"><i><font color="Green" title="E39">A1</font></i></a>, <a class="txt" href="hilbert3.html#E5:55_2"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="hilbert3.html#T24" target="_self" title="HILBERT3:th.24">Th24</a></i>;<br/><b>set </b><font color="Maroon" title="c12">M</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">h</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span>;<br/><a NAME="E10:55_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">h</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">M</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">h</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> as   <a href="pboole.html#NM2" title="PBOOLE:NM.2">ManySortedFunction</a> of  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">f</font></span>)</span> <b>by </b><i><a class="txt" href="hilbert3.html#E7:55_2"><i><font color="Green" title="E44">A6</font></i></a>, <a class="ref" href="pboole.html#D3" title="PBOOLE:def.3">PBOOLE:def 3</a></i>;<br/><a NAME="E12:55_2"/><i><font color="Green" title="E45">A7</font></i>: 
for <font color="Olive" title="b1">g</font> being  <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>  st <font color="Olive" title="b1">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">f</font></span>)</span> holds <br/><font color="Maroon" title="c13">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">f</font> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Olive" title="b1">g</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">g</font> be   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a>;<br/>

<b>assume </b><a NAME="E1:55_2_2"/><i><font color="Green" title="E46">A8</font></i>: 
<font color="Maroon" title="c14">g</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <font color="Maroon" title="c7">f</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c15">G</font> = <font color="Maroon" title="c14">g</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font>, <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c2">q</font> <b>by </b><i><a class="txt" href="hilbert3.html#E6:55_2"><i><font color="Green" title="E43">A5</font></i></a>, <a class="txt" href="hilbert3.html#E1:55_2_2"><i><font color="Green" title="E46">A8</font></i></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">FF</font> = <font color="Maroon" title="c9">ff</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c2">q</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c3">r</font></span>)</span> <b>by </b><i><a class="txt" href="hilbert3.html#E5:55_2"><i><font color="Green" title="E42">A4</font></i></a></i>;<br/>
<a NAME="E4:55_2_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c16">FF</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:55_2_2"/><b>then </b><i><font color="Green" title="E47">A9</font></i>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c15">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font>
 
<b>by </b><i><a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</a></i>;<br/>
<a NAME="E6:55_2_2"/><i><font color="Green" title="E48">A10</font></i>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c15">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T20" target="_self" title="HILBERT3:th.20">Th20</a></i>;<br/>
<a NAME="E7:55_2_2"/><b>then </b>
<font color="Maroon" title="c16">FF</font> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c15">G</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font>, <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c3">r</font>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E5:55_2_2"><i><font color="Green" title="E47">A9</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E8:55_2_2"/><b>then </b>
<font color="Maroon" title="c16">FF</font> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c15">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c3">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<a NAME="E9:55_2_2"/><b>then </b><i><font color="Green" title="E49">A11</font></i>: 
<font color="Maroon" title="c16">FF</font> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c15">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">GG</font> = <font color="Maroon" title="c16">FF</font> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c15">G</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font>, <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c3">r</font> <b>by </b><i><a class="txt" href="hilbert3.html#E5:55_2_2"><i><font color="Green" title="E47">A9</font></i></a>, <a class="txt" href="hilbert3.html#E6:55_2_2"><i><font color="Green" title="E48">A10</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E11:55_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span>
 
;<br/>
<a NAME="E12:55_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/>
<a NAME="E13:55_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font>, <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c2">q</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c3">r</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E5:55_2"><i><font color="Green" title="E42">A4</font></i></a>, <a class="ref" href="funct_2.html#T121" title="FUNCT_2:th.121">FUNCT_2:121</a></i>;<br/>
<b>then </b><i><font color="Green" title="E50">A12</font></i>:  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span> = 
 <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c2">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hilbert3.html#T6" target="_self" title="HILBERT3:th.6">Th6</a></i>
<br/>.= 
 <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c2">q</font></span>)</span>
<b>by </b><i><a class="ref" href="card_3.html#T20" title="CARD_3:th.20">CARD_3:20</a></i>
<br/>.= 
 <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span>
<b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>
;<br/>
<a NAME="E15:55_2_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">q</font></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c15">G</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c1">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<font color="Maroon" title="c2">q</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11" title="FUNCT_2:th.11">FUNCT_2:11</a></i>;<br/>
<a NAME="E16:55_2_2"/><b>then </b><i><font color="Green" title="E51">A13</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">q</font></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c1">p</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="card_3.html#K4" title="CARD_3:func.4">product</a> <span class="p1">(<span class="default"><a href="funct_6.html#K2" title="FUNCT_6:func.2">doms</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E14:55_2_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/>
<a NAME="E17:55_2_2"/><b>then </b><i><font color="Green" title="E52">A14</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="hilbert3.html#E14:55_2_2"><i><font color="Green" title="E50">A12</font></i></a>, <a class="ref" href="hilbert3.html#T38" target="_self" title="HILBERT3:th.38">Th38</a></i>;<br/>
<i><font color="Green" title="E53">A15</font></i>: <font color="Maroon" title="c8">X</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> 
<b>by </b><i><a class="ref" href="hilbert3.html#T36" target="_self" title="HILBERT3:th.36">Th36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">q</font></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="hilbert3.html#T27" target="_self" title="HILBERT3:th.27">Th27</a></i>
;<br/>
<i><font color="Green" title="E54">A16</font></i>: <font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">g</font></span>)</span> = 
<font color="Maroon" title="c11">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">q</font></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c15">G</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hilbert3.html#T38" target="_self" title="HILBERT3:th.38">Th38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font></span>)</span> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">q</font></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E16:55_2_2"><i><font color="Green" title="E51">A13</font></i></a>, <a class="ref" href="pralg_2.html#D8" title="PRALG_2:def.8">PRALG_2:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">X</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">ff</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">q</font></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c14">g</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hilbert3.html#T38" target="_self" title="HILBERT3:th.38">Th38</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">X</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c16">FF</font></span>)</span> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c2">q</font></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <font color="Maroon" title="c14">g</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c1">p</font></span>)</span>
<b>by </b><i><a class="ref" href="hilbert3.html#T18" target="_self" title="HILBERT3:th.18">Th18</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c3">r</font></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">FF</font> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c15">G</font></span>)</span></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<font color="Maroon" title="c1">p</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E18:55_2_2"><i><font color="Green" title="E53">A15</font></i></a>, <a class="ref" href="hilbert3.html#T28" target="_self" title="HILBERT3:th.28">Th28</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">GG</font>
<b>by </b><i><a class="ref" href="hilbert3.html#T38" target="_self" title="HILBERT3:th.38">Th38</a></i>
;<br/>
<b>thus </b><font color="Maroon" title="c13">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">g</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">h</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">g</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E7:55_2"><i><font color="Green" title="E44">A6</font></i></a>, <a class="txt" href="hilbert3.html#E1:55_2_2"><i><font color="Green" title="E46">A8</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">GG</font></span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E17:55_2_2"><i><font color="Green" title="E52">A14</font></i></a>, <a class="txt" href="hilbert3.html#E19:55_2_2"><i><font color="Green" title="E54">A16</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">GG</font>
<b>by </b><i><a class="txt" href="hilbert3.html#E9:55_2_2"><i><font color="Green" title="E49">A11</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p2">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c4">V</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">GG</font>
<b>by </b><i><a class="ref" href="funct_2.html#T88" title="FUNCT_2:th.88">FUNCT_2:88</a></i>
<br/>.= 
<font color="Maroon" title="c7">f</font> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c14">g</font>
<b>by </b><i><a class="txt" href="hilbert3.html#E9:55_2_2"><i><font color="Green" title="E49">A11</font></i></a>, <a class="ref" href="funct_1.html#T35" title="FUNCT_1:th.35">FUNCT_1:35</a></i>
;<br/>


</div><b>end;</b></div><b>thus </b><font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font> = 
 <a href="pralg_2.html#K2" title="PRALG_2:func.2">Frege</a> <font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="txt" href="hilbert3.html#E4:55"><i><font color="Green" title="E41">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c11">h</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E12:55_2"><i><font color="Green" title="E45">A7</font></i></a>, <a class="ref" href="pralg_2.html#D8" title="PRALG_2:def.8">PRALG_2:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">h</font>
<b>by </b><i><a class="ref" href="hilbert3.html#T37" target="_self" title="HILBERT3:th.37">Th37</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p0">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E4:55"><i><font color="Green" title="E41">A3</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p5">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">f</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><font color="Maroon" title="c5">F</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c5">F</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c6">P</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c2">q</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <span class="p4">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c3">r</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">F</font>
<b>by </b><i><a class="ref" href="hilbert3.html#T37" target="_self" title="HILBERT3:th.37">Th37</a></i>
;<br/><br/>


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