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<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> ,   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Maroon">a<sub>3</sub></font>,<font color="Maroon">a<sub>4</sub></font>);<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ,   <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Maroon">a<sub>3</sub></font>,<font color="Maroon">a<sub>4</sub></font>);<br/>
<a NAME="E1:40_1"/><i><font color="Green">E29</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> <br/> for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   ex <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>]
 
;<br/>
<a NAME="E2:40_1"/><i><font color="Green">E30</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> <br/> for <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>   ex <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">d</font>]
 
;<br/>
<a NAME="E3:40_1"/><i><font color="Green">E33</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> <br/> for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">d</font>] holds <br/><font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">d</font>
 
;<br/>
<a NAME="E4:40_1"/><i><font color="Green">E34</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> <br/> for <font color="Olive">a</font>, <font color="Olive">b</font>, <font color="Olive">c</font>, <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">c</font>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Olive">a</font>,<font color="Olive">b</font>,<font color="Olive">d</font>] holds <br/><font color="Olive">c</font> <a href="hidden.html#R1">=</a> <font color="Olive">d</font>
 
;<br/>
<b>consider </b><font color="Maroon">M</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="hilbert1.html#K1">HP-WFF</a> <b> such that </b><br/><a NAME="E6:40_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>and </b><br/><a NAME="E7:40_1"/><i><font color="Green">E37</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>)
 <b>and </b><br/><a NAME="E8:40_1"/><i><font color="Green">E38</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">S<sub>1</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font>,<font color="Maroon">M</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>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">p</font>,<font color="Olive">q</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Olive">q</font>,<font color="Maroon">M</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>] )
 <b>from </b><i><a class="ref" href="hilbert2.html#S3" target="_self">HILBERT2:sch 3</a>(, , <a class="txt" href="hilbert2.html#E3:40_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="hilbert2.html#E4:40_1"><i><font color="Green">E34</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon">M</font>
;<br/>

<b>thus </b><a NAME="E9:40_1"/>
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>()
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E10:40_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>(<font color="Olive">n</font>)
 <b>by </b><i><a class="txt" href="hilbert2.html#E6:40_1"><i><font color="Green">E36</font></i></a></i>;<br/>

<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>set </b><font color="Maroon">x</font> = <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>;<br/>
<b>set </b><font color="Maroon">y</font> = <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>;<br/>
<b>thus </b><a NAME="E11:40_1"/>
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font></span>)</span>)
 <b>by </b><i><a class="txt" href="hilbert2.html#E7:40_1"><i><font color="Green">E37</font></i></a></i>;<br/>

<b>thus </b><a NAME="E12:40_1"/>
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font></span>)</span>)
 <b>by </b><i><a class="txt" href="hilbert2.html#E7:40_1"><i><font color="Green">E37</font></i></a></i>;<br/>


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