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

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/><b>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/><b>let </b><font color="Maroon">V</font> be   <a href="hilbert3.html#NM1">SetValuation</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="hilbert3.html#D7" target="_self">HILBERT3:def 7</a><br/>





<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="pboole.html#M1">ManySortedSet</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font> = <span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:54"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>) <a href="hidden.html#R2">in</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">p</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font>;<br/>

<a NAME="E1:54_1"/>
<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font> is   <a href="funct_2.html#NM1">Function</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font>, <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/>
<a NAME="E2:54_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<b>hence </b><a NAME="E3:54_1"/>
<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">p</font></span>)</span>
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font>, <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">p</font></span>)</span><b> such that </b><br/><a NAME="E3:54"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S8">FUNCT_2:sch 8</a>(<a class="ref" href="hilbert3.html#T6" target="_self">Th6</a>);<br/></i>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>let </b><font color="Maroon">P</font> be    <a href="hilbert3.html#M1" target="_self">Permutation</a> of <font color="Maroon">V</font>;<br/>

<a NAME="E4:54"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="fraenkel.html#K1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#T11">FUNCT_2:11</a></i>;<br/>
<a NAME="E5:54"/><b>then </b>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">p</font></span>)</span></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:54"/>
<font color="Maroon">f</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="abian.html#D3">ABIAN:def 3</a><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font>;<br/><a NAME="E1:54_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font>
 ;<br/><a NAME="E2:54_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="funct_2.html#D3">FUNCT_2:def 3</a></i>;<br/><a NAME="E3:54_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="partfun2.html#T79">PARTFUN2:79</a></i>;<br/><b>reconsider </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> as   <a href="funct_2.html#NM1">Function</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font>, <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font> <b>by </b><i><a class="ref" href="funcop_1.html#T57">FUNCOP_1:57</a></i>;<br/><b>thus </b><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> = 
<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <font color="Maroon">x</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="partfun1.html#K6">id</a> <span class="p3">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T34">FUNCT_1:34</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T88">FUNCT_2:88</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i>, <a class="ref" href="funcop_1.html#T23">FUNCOP_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_1.html#NK3" target="_self">*</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" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">q</font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="relat_1.html#K10">"</a> <span class="p3">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font></span>)</span></span>)</span> <a href="pre_circ.html#K2">--&gt;</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T48">FUNCT_2:48</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">g</font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">q</font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T25">FUNCOP_1:25</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">g</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">q</font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">g</font>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<span class="p4">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font>
<b>by </b><i><a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><font color="Maroon">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">p</font></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f</font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<font color="Maroon">p</font></span>)</span> <a href="funct_2.html#K2">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">P</font>,<span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">p</font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">f</font>
<b>by </b><i/>
;<br/><br/>


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