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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_6.html#K7" title="FUNCT_6:func.7">Frege</a> <font color="Maroon" title="c1">p</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c4">V</font> be   <a href="hilbert3.html#NM1" title="HILBERT3:NM.1">SetValuation</a>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="hilbert3.html#D7" target="_self" title="HILBERT3:def.7">HILBERT3:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  st <br/> for <font color="Olive" title="b2">P</font> being    <a href="hilbert3.html#M1" title="HILBERT3:mode.1">Permutation</a> of <font color="Maroon" title="c4">V</font> holds  <font color="Olive" title="b1">x</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a>  <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> (<font color="Olive" title="b2">P</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>)</span><br/>

<a NAME="E1:52_1_1"/><span class="lab"><font color="Green" title="E37">A1</font></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="c2">q</font></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">(<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="funct_2.html#K9" title="FUNCT_2:func.9">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>) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></span>;<br/>
<a NAME="E2:52_1_1"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
 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="52_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">H<sub>1</sub></font>(<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>)</span><br/>

<a NAME="E1:52_1_1_1"/>
<font color="Maroon" title="c5">x</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></span>;<br/>
<a NAME="E2:52_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">x</font> is    <a href="funct_2.html#M2" title="FUNCT_2:mode.2">Element</a> of  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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="funct_2.html#K9" title="FUNCT_2:func.9">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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></span>;<br/>
<a NAME="E3:52_1_1_1"/><span class="kw">then </span>
 <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 <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="ref" href="hilbert3.html#T24" target="_self" title="HILBERT3:th.24">Th24</a></span>;<br/>
<a NAME="E4:52_1_1_1"/><span class="kw">then </span>
 <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="funct_2.html#K9" title="FUNCT_2:func.9">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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:52_1_1_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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">consider </span><font color="Maroon" title="c5">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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><span class="kw"> such that </span><br/><a NAME="E4:52_1_1"/><span class="lab"><font color="Green" title="E39">A3</font></span>: 
 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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S8" title="FUNCT_2:sch.8">FUNCT_2:sch 8</a>(<a class="txt" href="#E2:52_1_1"><span class="lab"><font color="Green" title="E38">A2</font></span></a>);<br/></span>
<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="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being    <a href="hilbert3.html#M1" title="HILBERT3:mode.1">Permutation</a> of <font color="Maroon" title="c4">V</font> holds  <font color="Maroon" title="c5">F</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a>  <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> (<font color="Olive" title="b1">P</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>)</span><br/>

<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">F</font> <a href="abian.html#R1" title="ABIAN:pred.1">is_a_fixpoint_of</a>  <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> (<font color="Maroon" title="c6">P</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>)</span><br/>

<a NAME="E5:52_1_1"/>
<font color="Maroon" title="c5">F</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></span>;<br/>
<a NAME="E6:52_1_1"/><span class="kw">then </span>
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:52_1_1"/>
<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>
 <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>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="abian.html#D3" title="ABIAN:def.3">ABIAN:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.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>,<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></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="52_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c7">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 <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>,<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> ;<br/><span class="kw">let </span><font color="Maroon" title="c8">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>); <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">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"><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="c8">f</font></span><br/><span class="kw">set </span><font color="Maroon" title="c9">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="c8">f</font>;<br/><a NAME="E2:52_1_1_2"/><span class="lab"><font color="Green" title="E40">A4</font></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>) <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">(<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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></span>;<br/><a NAME="E3:52_1_1_2"/>
<font color="Maroon" title="c8">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="E4:52_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">ff</font> = <font color="Maroon" title="c8">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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> <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T66" title="FUNCT_2:th.66">FUNCT_2:66</a></span>;<br/><a NAME="E6:52_1_1_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="c8">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="c10">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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#T38" target="_self" title="HILBERT3:th.38">Th38</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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="c8">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 <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:52_1_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <a class="txt" href="#E2:52_1_1_2"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="hilbert3.html#T24" target="_self" title="HILBERT3:th.24">Th24</a></span>;<br/><span class="kw">set </span><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/><span class="lab"><font color="Green" title="E41">A5</font></span>:  <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="c10">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="funcop_1.html#K7" title="FUNCOP_1:func.7">--&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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_1_1_2"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="hilbert3.html#T6" target="_self" title="HILBERT3:th.6">Th6</a></span>
<br/>.= 
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T11" title="CARD_3:th.11">CARD_3:11</a></span>
;<br/><a NAME="E9:52_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A6</font></span>: 
 <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="c10">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>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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="c8">f</font></span>)</span> ;<br/><a NAME="E11:52_1_1_2"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
 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="c8">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="c8">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="52_1_1_2_2"><span class="kw">proof </span></a><div class="add">

<a NAME="E1:52_1_1_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="c8">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="E2:52_1_1_2_2"/><span class="kw">then </span>
<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="c8">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></span>;<br/>
<a NAME="E3:52_1_1_2_2"/><span class="kw">then </span>
<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="c8">f</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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="funct_2.html#K9" title="FUNCT_2:func.9">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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:52_1_1_2"><span class="lab"><font color="Green" title="E40">A4</font></span></a>, <a class="ref" href="funct_2.html#T66" title="FUNCT_2:th.66">FUNCT_2:66</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E44">A8</font></span>:  <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="c8">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="funcop_1.html#K7" title="FUNCOP_1:func.7">--&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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#T6" target="_self" title="HILBERT3:th.6">Th6</a></span>
<br/>.= 
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_3.html#T11" title="CARD_3:th.11">CARD_3:11</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></span>
;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c14">FF</font> = <font color="Maroon" title="c10">ff</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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="funct_2.html#K9" title="FUNCT_2:func.9">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> <span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c15">g</font> be   <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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">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="c8">f</font></span>)</span> implies <font color="Maroon" title="c13">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c15">g</font> )</span><br/>

<span class="kw">assume </span><a NAME="E6:52_1_1_2_2"/><span class="lab"><font color="Green" title="E45">A9</font></span>: 
<font color="Maroon" title="c15">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="c8">f</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c13">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">g</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c15">g</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c16">G</font> = <font color="Maroon" title="c15">g</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:52_1_1_2"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E6:52_1_1_2_2"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="ref" href="funct_2.html#T66" title="FUNCT_2:th.66">FUNCT_2:66</a></span>;<br/>
<a NAME="E8:52_1_1_2_2"/>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">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>)
 
<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/>
<a NAME="E9:52_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A10</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">FF</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c16">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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pralg_1.html#D17" title="PRALG_1:def.17">PRALG_1:def 17</a></span>;<br/>
<a NAME="E10:52_1_1_2_2"/><span class="lab"><font color="Green" title="E47">A11</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">FF</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c16">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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#T21" target="_self" title="HILBERT3:th.21">Th20</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">GG</font> = <font color="Maroon" title="c14">FF</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c16">G</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:52_1_1_2_2"><span class="lab"><font color="Green" title="E46">A10</font></span></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#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/>
<a NAME="E12:52_1_1_2_2"/>
<font color="Maroon" title="c14">FF</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c16">G</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:52_1_1_2_2"><span class="lab"><font color="Green" title="E46">A10</font></span></a>, <a class="txt" href="#E10:52_1_1_2_2"><span class="lab"><font color="Green" title="E47">A11</font></span></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#T4" title="RELSET_1:th.4">RELSET_1:4</a></span>;<br/>
<a NAME="E13:52_1_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c14">FF</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c16">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="funct_2.html#K9" title="FUNCT_2:func.9">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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></span>;<br/>
<a NAME="E14:52_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A12</font></span>: 
<font color="Maroon" title="c14">FF</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c16">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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></span>;<br/>
<a NAME="E15:52_1_1_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="c16">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="funct_2.html#K9" title="FUNCT_2:func.9">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>)
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T8" title="FUNCT_2:th.8">FUNCT_2:8</a></span>;<br/>
<a NAME="E16:52_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A13</font></span>: 
<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="c15">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="c8">f</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1_1_2_2"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></span>;<br/>
<a NAME="E17:52_1_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A14</font></span>: 
<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="c16">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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1_1_2_2"><span class="lab"><font color="Green" title="E44">A8</font></span></a>, <a class="ref" href="hilbert3.html#T38" target="_self" title="HILBERT3:th.38">Th38</a></span>;<br/>
<span class="lab"><font color="Green" title="E51">A15</font></span>: <font color="Maroon" title="c7">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> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#T36" target="_self" title="HILBERT3:th.36">Th36</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#T27" target="_self" title="HILBERT3:th.27">Th27</a></span>
;<br/>
<span class="lab"><font color="Green" title="E52">A16</font></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"><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="c15">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="c16">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#T38" target="_self" title="HILBERT3:th.38">Th38</a></span>
<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="c8">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="c15">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:52_1_1_2_2"><span class="lab"><font color="Green" title="E49">A13</font></span></a>, <a class="ref" href="pralg_2.html#D2" title="PRALG_2:def.2">PRALG_2:def 2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c10">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="c15">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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#T38" target="_self" title="HILBERT3:th.38">Th38</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">X</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c14">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="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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#T19" target="_self" title="HILBERT3:th.19">Th18</a></span>
<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="c14">FF</font> <a href="pralg_1.html#K16" title="PRALG_1:func.16">..</a> <font color="Maroon" title="c16">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:52_1_1_2_2"><span class="lab"><font color="Green" title="E51">A15</font></span></a>, <a class="ref" href="hilbert3.html#T28" target="_self" title="HILBERT3:th.28">Th28</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#T38" target="_self" title="HILBERT3:th.38">Th38</a></span>
;<br/>
<span class="kw">thus </span><font color="Maroon" title="c13">M</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">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="c15">g</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:52_1_1_2"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E6:52_1_1_2_2"><span class="lab"><font color="Green" title="E45">A9</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:52_1_1_2_2"><span class="lab"><font color="Green" title="E50">A14</font></span></a>, <a class="txt" href="#E19:52_1_1_2_2"><span class="lab"><font color="Green" title="E52">A16</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:52_1_1_2_2"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T61" title="FUNCT_2:th.61">FUNCT_2:61</a></span>
<br/>.= 
<font color="Maroon" title="c8">f</font> <a href="pralg_1.html#K15" title="PRALG_1:func.15">..</a> <font color="Maroon" title="c15">g</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:52_1_1_2_2"><span class="lab"><font color="Green" title="E48">A12</font></span></a>, <a class="ref" href="funct_1.html#T18" title="FUNCT_1:th.18">FUNCT_1:18</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><span class="kw">thus </span><font color="Maroon" title="c5">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">f</font> = 
 <a href="pralg_2.html#K2" title="PRALG_2:func.2">Frege</a> <font color="Maroon" title="c8">f</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:52_1_1_2"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="ref" href="pralg_2.html#D2" title="PRALG_2:def.2">PRALG_2:def 2</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#T37" target="_self" title="HILBERT3:th.37">Th37</a></span>
<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="c8">f</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:52_1_1"><span class="lab"><font color="Green" title="E39">A3</font></span></a></span>
<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="c8">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
<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="c8">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T15" title="FUNCT_2:th.15">FUNCT_2:15</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T63" title="FUNCT_2:th.63">FUNCT_2:63</a></span>
<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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#T37" target="_self" title="HILBERT3:th.37">Th37</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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