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<span class="kw">let </span><font color="Maroon" title="c3">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="c3">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#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span>)</span><br/>

<span class="kw">take </span><font color="Maroon" title="c4">f</font> =  <a href="funct_3.html#K10" title="FUNCT_3:func.10">pr2</a> (<span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> (<font color="Maroon" title="c3">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="c3">V</font>,<font color="Maroon" title="c2">q</font>)</span>)</span>); <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="c3">V</font> holds  <font color="Maroon" title="c4">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#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span>)</span><br/>

<span class="kw">let </span><font color="Maroon" title="c5">P</font> be    <a href="hilbert3.html#M1" title="HILBERT3:mode.1">Permutation</a> of <font color="Maroon" title="c3">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="c4">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="c5">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#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span>)</span><br/>

<span class="lab"><font color="Green" title="E37">A1</font></span>:  <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="c5">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#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</font></span>)</span>)</span>)</span> = 
 <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> (<font color="Maroon" title="c3">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#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <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="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</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="c3">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</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="c3">V</font>,<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/>.= 
 <a href="funct_2.html#K9" title="FUNCT_2:func.9">Funcs</a> (<span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>,<span class="p2">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> (<font color="Maroon" title="c3">V</font>,<font color="Maroon" title="c2">q</font>)</span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> (<font color="Maroon" title="c3">V</font>,<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">hence </span><a NAME="E2:51_3_1"/>
<font color="Maroon" title="c4">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="c5">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#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c2">q</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>; <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="c4">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="c5">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#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">f</font></span><br/>

<a NAME="E3:51_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">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="c3">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</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="c3">V</font>,<font color="Maroon" title="c2">q</font>)</span>)</span>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:51_3_1"><span class="lab"><font color="Green" title="E37">A1</font></span></a>, <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="c6">F</font> = <font color="Maroon" title="c4">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="c3">V</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</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="c3">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#T66" title="FUNCT_2:th.66">FUNCT_2:66</a></span>;<br/>
<span class="kw">thus </span><span class="p1">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> (<font color="Maroon" title="c5">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#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> <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_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">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="c5">P</font>,<font color="Maroon" title="c2">q</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">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="c5">P</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</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="ref" href="hilbert3.html#T37" target="_self" title="HILBERT3:th.37">Th37</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="c5">P</font>,<font color="Maroon" title="c2">q</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="funct_1.html#NK3" title="FUNCT_1:NK.3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> (<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c1">p</font>)</span>)</span>,<span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> (<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">q</font>)</span>)</span></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#T35" target="_self" title="HILBERT3:th.35">Th35</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="c5">P</font>,<font color="Maroon" title="c2">q</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><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="c5">P</font>,<font color="Maroon" title="c1">p</font>)</span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span>,<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="c5">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></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="functor0.html#T6" title="FUNCTOR0:th.6">FUNCTOR0:6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> (<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">q</font>)</span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><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="c5">P</font>,<font color="Maroon" title="c1">p</font>)</span>)</span> <a href="funct_2.html#K2" title="FUNCT_2:func.2">"</a></span>)</span>,<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="c5">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></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" title="RELAT_1:th.36">RELAT_1:36</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> (<font color="Maroon" title="c5">P</font>,<font color="Maroon" title="c2">q</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"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> (<font color="Maroon" title="c5">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="c6">F</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="hilbert3.html#T17" target="_self" title="HILBERT3:th.17">Th16</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="c5">P</font>,<font color="Maroon" title="c2">q</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="c5">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></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T36" title="RELAT_1:th.36">RELAT_1:36</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="c3">V</font>,<font color="Maroon" title="c2">q</font>)</span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c6">F</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="c4">f</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T17" title="FUNCT_2:th.17">FUNCT_2:17</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>
