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<div class="add">

<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>3</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>4</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">a<sub>2</sub></font> implies  ex <font color="Olive">p'</font> being  <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">a<sub>1</sub></font> ex <font color="Olive">q'</font> being  <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">a<sub>2</sub></font> st <br/>( <font color="Olive">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> &amp; <font color="Olive">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>4</sub></font> &amp; <font color="Maroon">a<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Olive">p'</font>,<font color="Olive">q'</font></span><a href="funct_3.html#K16">:]</a></span> ) ) &amp; ( (  not <font color="Maroon">a<sub>3</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">a<sub>1</sub></font> or  not <font color="Maroon">a<sub>4</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">a<sub>2</sub></font> ) implies <font color="Maroon">a<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) );<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>3</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>4</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">a<sub>2</sub></font> implies  ex <font color="Olive">p'</font> being  <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">a<sub>1</sub></font> ex <font color="Olive">q'</font> being  <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">a<sub>2</sub></font> st <br/>( <font color="Olive">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>3</sub></font> &amp; <font color="Olive">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>4</sub></font> &amp; <font color="Maroon">a<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">p'</font> <a href="hilbert3.html#K1" target="_self">=&gt;</a> <font color="Olive">q'</font> ) ) &amp; ( (  not <font color="Maroon">a<sub>3</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">a<sub>1</sub></font> or  not <font color="Maroon">a<sub>4</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">a<sub>2</sub></font> ) implies <font color="Maroon">a<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  ) );<br/>
<a NAME="E1:42_1_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> 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>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>proof </b></a><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">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1_1_1"/>
( ( <font color="Maroon">a</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">b</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font> ) or  not <font color="Maroon">a</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font> or  not <font color="Maroon">b</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:42_1_1_1_1_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">a</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E2:42_1_1_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">b</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">a</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <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="txt" href="hilbert3.html#E1:42_1_1"><i><font color="Green">E33</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">b</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font> <b>by </b><i/>;<br/><b>take </b>
<span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">p'</font>,<font color="Maroon">q'</font></span><a href="funct_3.html#K16">:]</a></span>
;<br/><b>thus </b><a NAME="E5:42_1_1_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>,<span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">p'</font>,<font color="Maroon">q'</font></span><a href="funct_3.html#K16">:]</a></span>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_1_1_2"/>
(  not <font color="Maroon">a</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font> or  not <font color="Maroon">b</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_1_1_2"/>
 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="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">c</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E2:42_1_1"/><i><font color="Green">E37</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>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_2"><b>proof </b></a><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">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be    <a href="hidden.html#M1">set</a> ;<br/>





<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1_2_1"/>
( ( <font color="Maroon">a</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font> &amp; <font color="Maroon">b</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font> ) or  not <font color="Maroon">a</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font> or  not <font color="Maroon">b</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_2_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:42_1_1_2_1_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">a</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E2:42_1_1_2_1_1"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">b</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font>
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">a</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font> <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">b</font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font> <b>by </b><i/>;<br/><b>take </b>
<font color="Maroon">p'</font> <a href="hilbert3.html#K1" target="_self">=&gt;</a> <font color="Maroon">q'</font>
;<br/><b>thus </b><a NAME="E5:42_1_1_2_1_1"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Maroon">p'</font> <a href="hilbert3.html#K1" target="_self">=&gt;</a> <font color="Maroon">q'</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_2_1_2"/>
(  not <font color="Maroon">a</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font> or  not <font color="Maroon">b</font> is   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_1_2_1_2"/>
 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="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">a</font>,<font color="Maroon">b</font>,<font color="Olive">d</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<a NAME="E3:42_1_1"/><i><font color="Green">E74</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="E4:42_1_1"/><i><font color="Green">E81</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>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></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="E6:42_1_1"/><i><font color="Green">E103</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>  <a href="partfun1.html#K6">id</a> 1
 <b>and </b><br/><a NAME="E7:42_1_1"/><i><font color="Green">E104</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">M</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">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>and </b><br/><a NAME="E8:42_1_1"/><i><font color="Green">E105</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">HILBERT2:sch 3</a>(, <a class="txt" href="hilbert3.html#E1:42_1_1"><i><font color="Green">E33</font></i></a>, , );<br/></i>
<b>take </b>
<font color="Maroon">M</font>
;<br/>

<b>defpred </b><font color="Maroon">S<sub>3</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">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> is   <a href="funct_2.html#NM2">Permutation</a> of <span class="p1">(<span class="default"><a href="hilbert3.html#K2" target="_self">SetVal</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E9:42_1_1"/>
<span class="p1">(<span class="default"><a href="hilbert3.html#K2" target="_self">SetVal</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="hilbert1.html#K2">VERUM</a>  <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i/>;<br/>
<a NAME="E10:42_1_1"/><b>then </b><i><font color="Green">E106</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="hilbert1.html#K2">VERUM</a> ]
 
<b>by </b><i/>;<br/>
<a NAME="E11:42_1_1"/><i><font color="Green">E107</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>3</sub></font>[ <a href="hilbert2.html#K1">prop</a> <font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_3"><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/>

<a NAME="E1:42_1_1_3"/>
( <font color="Maroon">M</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> <a href="hidden.html#R1">=</a> <font color="Maroon">P</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <span class="p1">(<span class="default"><a href="hilbert3.html#K2" target="_self">SetVal</a> <font color="Maroon">V</font></span>)</span> <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> <a href="hidden.html#R1">=</a> <font color="Maroon">V</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="txt" href="hilbert3.html#E2:42_1_1"><i><font color="Green">E37</font></i></a>, </i>;<br/>
<b>hence </b><a NAME="E2:42_1_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[ <a href="hilbert2.html#K1">prop</a> <font color="Maroon">n</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E12:42_1_1"/><i><font color="Green">E118</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>3</sub></font>[<font color="Olive">r</font>] &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">s</font>] holds <br/>( <font color="Maroon">S<sub>3</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>3</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="42_1_1_4"><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><b>that </b><br/><a NAME="E1:42_1_1_4"/><i><font color="Green">E119</font></i>: 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> is   <a href="funct_2.html#NM2">Permutation</a> of <span class="p1">(<span class="default"><a href="hilbert3.html#K2" target="_self">SetVal</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>
 <b>and </b><br/><a NAME="E2:42_1_1_4"/><i><font color="Green">E120</font></i>: 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> is   <a href="funct_2.html#NM2">Permutation</a> of <span class="p1">(<span class="default"><a href="hilbert3.html#K2" target="_self">SetVal</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>
 ;<br/>

<b>consider </b><font color="Maroon">p'</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">r</font>, <font color="Maroon">q'</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">s</font><b> such that </b><br/><a NAME="E4:42_1_1_4"/><i><font color="Green">E121</font></i>: 
( <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">M</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> <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">p'</font>,<font color="Maroon">q'</font></span><a href="funct_3.html#K16">:]</a></span> )
 <b>by </b><i>, <a class="ref" href="hilbert3.html#T5" target="_self">Th5</a>, <a class="ref" href="hilbert3.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E5:42_1_1_4"/>
<span class="p1">(<span class="default"><a href="hilbert3.html#K2" target="_self">SetVal</a> <font color="Maroon">V</font></span>)</span> <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> <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"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">s</font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:42_1_1_4"/>
<font color="Maroon">M</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> is   <a href="funct_2.html#NM2">Permutation</a> of <span class="p1">(<span class="default"><a href="hilbert3.html#K2" target="_self">SetVal</a> <font color="Maroon">V</font></span>)</span> <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>, </i>;<br/>

<b>consider </b><font color="Maroon">p'</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">r</font>, <font color="Maroon">q'</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">s</font><b> such that </b><br/><a NAME="E8:42_1_1_4"/><i><font color="Green">E122</font></i>: 
( <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> &amp; <font color="Maroon">M</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="hidden.html#R1">=</a> <font color="Maroon">p'</font> <a href="hilbert3.html#K1" target="_self">=&gt;</a> <font color="Maroon">q'</font> )
 <b>by </b><i>, <a class="ref" href="hilbert3.html#T5" target="_self">Th5</a>, <a class="ref" href="hilbert3.html#T6" target="_self">Th6</a></i>;<br/>
<a NAME="E9:42_1_1_4"/>
<span class="p1">(<span class="default"><a href="hilbert3.html#K2" target="_self">SetVal</a> <font color="Maroon">V</font></span>)</span> <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="hidden.html#R1">=</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">r</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">s</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E10:42_1_1_4"/>
<font color="Maroon">M</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> is   <a href="funct_2.html#NM2">Permutation</a> of <span class="p1">(<span class="default"><a href="hilbert3.html#K2" target="_self">SetVal</a> <font color="Maroon">V</font></span>)</span> <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="E13:42_1_1"/><i><font color="Green">E123</font></i>: 
for <font color="Olive">p</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>3</sub></font>[<font color="Olive">p</font>]
 
<b>from </b><i><a class="ref" href="hilbert2.html#S2">HILBERT2:sch 2</a>(, , <a class="ref" href="hilbert3.html#T4" target="_self">Th4</a>);<br/></i>
<b>thus </b><a NAME="E14:42_1_1"/>
<font color="Maroon">M</font> is    <a href="pboole.html#M3">ManySortedFunction</a> of  <a href="hilbert3.html#K2" target="_self">SetVal</a> <font color="Maroon">V</font>, <a href="hilbert3.html#K2" target="_self">SetVal</a> <font color="Maroon">V</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="pboole.html#D18">PBOOLE:def 18</a><br/>

<b>thus </b><a NAME="E1:42_1_1_5"/>
(  not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="hilbert1.html#K1">HP-WFF</a>  or <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> is   <a href="relset_1.html#M2">M6</a>(<span class="p1">(<span class="default"><a href="hilbert3.html#K2" target="_self">SetVal</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>,<span class="p1">(<span class="default"><a href="hilbert3.html#K2" target="_self">SetVal</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>) )
 <b>by </b><i><a class="ref" href="hilbert3.html#T5" target="_self">Th5</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E15:42_1_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>  <a href="partfun1.html#K6">id</a> 1
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E16:42_1_1"/>
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">M</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">P</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>
 <b>by </b><i><a class="txt" href="hilbert3.html#E2:42_1_1"><i><font color="Green">E37</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/>



<a NAME="E17:42_1_1"/><i><font color="Green">E124</font></i>: 
( <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> is   <a href="funct_2.html#NM2">Permutation</a> of <span class="p1">(<span class="default"><a href="hilbert3.html#K2" target="_self">SetVal</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> is   <a href="funct_2.html#NM2">Permutation</a> of <span class="p1">(<span class="default"><a href="hilbert3.html#K2" target="_self">SetVal</a> <font color="Maroon">V</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="hilbert3.html#T5" target="_self">Th5</a></i>;<br/>
<b>then consider </b><font color="Maroon">p'</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font>, <font color="Maroon">q'</font> being   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font><b> such that </b><br/><a NAME="E19:42_1_1"/><i><font color="Green">E125</font></i>: 
( <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> &amp; <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> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">p'</font>,<font color="Maroon">q'</font></span><a href="funct_3.html#K16">:]</a></span> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">p'</font>
;<br/>

<b>take </b>
<font color="Maroon">q'</font>
;<br/>

<b>thus </b><a NAME="E20:42_1_1"/>
( <font color="Maroon">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> &amp; <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> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">p'</font>,<font color="Maroon">q'</font></span><a href="funct_3.html#K16">:]</a></span> )
 <b>by </b><i/>;<br/>

<a NAME="E21:42_1_1"/>
 ex <font color="Olive">p'</font> being  <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">p</font> ex <font color="Olive">q'</font> being  <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">V</font>,<font color="Maroon">q</font> st <br/>( <font color="Olive">p'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> &amp; <font color="Olive">q'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font> &amp; <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="Olive">p'</font> <a href="hilbert3.html#K1" target="_self">=&gt;</a> <font color="Olive">q'</font> )
 
<b>by </b><i>, <a class="ref" href="hilbert3.html#T6" target="_self">Th6</a></i>;<br/>
<b>hence </b><a NAME="E22:42_1_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">p'</font> <a href="hilbert3.html#K1" target="_self">=&gt;</a> <font color="Maroon">q'</font>
 <b>by </b><i/>;<br/>


</div>
