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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> ] means  <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">a<sub>1</sub></font> is <a href="funct_2.html#V3">bijective</a>;<br/>
<a NAME="E1:47_1_1"/>
 <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>2</sub></font>,<a href="hilbert1.html#K2">VERUM</a>  <a href="funct_2.html#R2">=</a>  <a href="partfun1.html#K6">id</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>1</sub></font>,<a href="hilbert1.html#K2">VERUM</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E2:47_1_1"/><b>then </b><i><font color="Green">E37</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert1.html#K2">VERUM</a> ]
 
;<br/>
<a NAME="E3:47_1_1"/><i><font color="Green">E38</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert2.html#K1">prop</a> <font color="Olive">b<sub>1</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:47_1_1_1"/><i><font color="Green">E39</font></i>: 
 <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T30" target="_self">Th30</a></i>;<br/>
<a NAME="E2:47_1_1_1"/>
 <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T34" target="_self">Th34</a></i>;<br/>
<b>hence </b><a NAME="E3:47_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert2.html#K1">prop</a> <font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="hilbert3.html#E1:47_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="hilbert3.html#D4" target="_self">Def4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:47_1_1"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></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">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">b<sub>2</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">b<sub>2</sub></font>] )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>



<a NAME="E1:47_1_1_2"/><i><font color="Green">E40</font></i>: 
 <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T31" target="_self">Th31</a></i>;<br/>
<a NAME="E2:47_1_1_2"/><i><font color="Green">E41</font></i>: 
 <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>1</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>5</sub></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">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T32" target="_self">Th32</a></i>;<br/>
<b>assume </b><a NAME="E3:47_1_1_2"/>
 <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> is <a href="funct_2.html#V3">bijective</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>4</sub></font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font> ;<br/>
<b>assume </b><a NAME="E5:47_1_1_2"/>
 <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> is <a href="funct_2.html#V3">bijective</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> =  <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font> as   <a href="funct_2.html#NM2">Permutation</a> of  <a href="hilbert3.html#K3" target="_self">SetVal</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> ;<br/>
<a NAME="E7:47_1_1_2"/>
 <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#R1">=</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>7</sub></font></span><a href="funct_3.html#K16">:]</a></span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T35" target="_self">Th35</a></i>;<br/>
<b>hence </b><a NAME="E8:47_1_1_2"/>
 <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> is <a href="funct_2.html#V3">bijective</a>
 <b>by </b><i><a class="txt" href="hilbert3.html#E1:47_1_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="hilbert3.html#T25" target="_self">Th25</a></i>;<br/>

<a NAME="E9:47_1_1_2"/>
 <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="funct_2.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="hilbert3.html#K1" target="_self">=&gt;</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T36" target="_self">Th36</a></i>;<br/>
<b>hence </b><a NAME="E10:47_1_1_2"/>
 <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> is <a href="funct_2.html#V3">bijective</a>
 <b>by </b><i><a class="txt" href="hilbert3.html#E2:47_1_1_2"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:47_1_1"/>
for <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="hilbert2.html#S2">HILBERT2:sch 2</a>(<a class="txt" href="hilbert3.html#E2:47_1_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="hilbert3.html#E3:47_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="hilbert3.html#E4:47_1_1"><i><font color="Green">E39</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:47_1_1"/>
 <a href="hilbert3.html#K5" target="_self">Perm</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> is <a href="funct_2.html#V3">bijective</a>
 ;<br/>


</div>
