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

<b>let </b><font color="Maroon">M1</font> be    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="hilbert1.html#K1">HP-WFF</a> , <font color="Maroon">M2</font> be    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:35_1_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E2:35_1_2"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>
 <b>and </b><br/><a NAME="E3:35_1_2"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  holds <br/> ( <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font></span>)</span> )
 <b>and </b><br/><a NAME="E4:35_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> 1
 <b>and </b><br/><a NAME="E5:35_1_2"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>
 <b>and </b><br/><a NAME="E6:35_1_2"/><i><font color="Green">E103</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  holds <br/> ( <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">q</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span> &amp; <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">q</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font></span>)</span> )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ] <b>means </b><font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E7:35_1_2"/><i><font color="Green">E104</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert1.html#K2">VERUM</a> ]
 
<b>by </b><i>, <a class="ref" href="hilbert3.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E8:35_1_2"/><i><font color="Green">E105</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert2.html#K1">prop</a> <font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>thus </b><font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> <font color="Maroon">n</font></span>)</span> = 
<font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font>
<b>by </b><i/>
<br/>.= 
<font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E9:35_1_2"/><i><font color="Green">E106</font></i>: 
for <font color="Olive">r</font>, <font color="Olive">s</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">s</font>] holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">s</font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">r</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">s</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:35_1_2_2"/><i><font color="Green">E107</font></i>: 
( <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> &amp; <font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> )
 ;<br/>

<b>thus </b><font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span> = 
<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
<b>by </b><i><a class="ref" href="hilbert3.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
<font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i/>
;<br/>

<b>thus </b><font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span> = 
 <a href="funct_2.html#K1">Funcs</a> <span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i><a class="ref" href="hilbert3.html#T2" target="_self">Th2</a>, </i>
<br/>.= 
<font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">s</font></span>)</span>
<b>by </b><i/>
;<br/>


</div><b>end;</b></div>
<a NAME="E10:35_1_2"/>
for <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">r</font>]
 
<b>from </b><i><a class="ref" href="hilbert2.html#S2">HILBERT2:sch 2</a>(, , );<br/></i>
<a NAME="E11:35_1_2"/><b>then </b>
for <font color="Olive">r</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="hilbert1.html#K1">HP-WFF</a>  holds <br/><font color="Maroon">M1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">r</font>
 
;<br/>
<b>hence </b><a NAME="E12:35_1_2"/>
<font color="Maroon">M1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">M2</font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
