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

<b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be    <a href="pboole.html#M3">ManySortedFunction</a> of  <a href="hilbert3.html#K2" target="_self">SetVal</a> <font color="Maroon">c<sub>1</sub></font>, <a href="hilbert3.html#K2" target="_self">SetVal</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:42_1_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.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="E2:42_1_2"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> <font color="Olive">b<sub>1</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="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E3:42_1_2"/><i><font color="Green">E34</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> ex <font color="Olive">b<sub>3</sub></font> being  <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="Olive">b<sub>1</sub></font>ex <font color="Olive">b<sub>4</sub></font> being  <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="Olive">b<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="funct_3.html#K16">:]</a></span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="hilbert3.html#K1" target="_self">=&gt;</a> <font color="Olive">b<sub>4</sub></font> )
 <b>and </b><br/><a NAME="E4:42_1_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.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="E5:42_1_2"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> <font color="Olive">b<sub>1</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="Olive">b<sub>1</sub></font>
 <b>and </b><br/><a NAME="E6:42_1_2"/><i><font color="Green">E37</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> ex <font color="Olive">b<sub>3</sub></font> being  <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="Olive">b<sub>1</sub></font>ex <font color="Olive">b<sub>4</sub></font> being  <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="Olive">b<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="funct_3.html#K16">[:</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font></span><a href="funct_3.html#K16">:]</a></span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Olive">b<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="hilbert3.html#K1" target="_self">=&gt;</a> <font color="Olive">b<sub>4</sub></font> )
 ;<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> ] means <font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E7:42_1_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert1.html#K2">VERUM</a> ]
 
<b>by </b><i><a class="txt" href="hilbert3.html#E1:42_1_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="hilbert3.html#E4:42_1_2"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E8:42_1_2"/><i><font color="Green">E39</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="42_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> = 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font>
<b>by </b><i><a class="txt" href="hilbert3.html#E2:42_1_2"><i><font color="Green">E33</font></i></a></i>
<br/>.= 
<font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1">prop</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="hilbert3.html#E5:42_1_2"><i><font color="Green">E36</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E9:42_1_2"/><i><font color="Green">E40</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="42_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></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="E1:42_1_2_2"/><i><font color="Green">E41</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <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>, <font color="Maroon">c<sub>8</sub></font> being   <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>6</sub></font><b> such that </b><br/><a NAME="E3:42_1_2_2"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>and </b><br/><a NAME="E4:42_1_2_2"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">c<sub>6</sub></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">c<sub>7</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="funct_3.html#K16">:]</a></span>
 <b>and </b><br/><a NAME="E5:42_1_2_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="hilbert3.html#K1" target="_self">=&gt;</a> <font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="txt" href="hilbert3.html#E3:42_1_2"><i><font color="Green">E34</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <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>, <font color="Maroon">c<sub>10</sub></font> being   <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>6</sub></font><b> such that </b><br/><a NAME="E7:42_1_2_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>and </b><br/><a NAME="E8:42_1_2_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">c<sub>6</sub></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">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font></span><a href="funct_3.html#K16">:]</a></span>
 <b>and </b><br/><a NAME="E9:42_1_2_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="hilbert3.html#K1" target="_self">=&gt;</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="hilbert3.html#E6:42_1_2"><i><font color="Green">E37</font></i></a></i>;<br/>
<b>thus </b><a NAME="E10:42_1_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbert3.html#E1:42_1_2_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="hilbert3.html#E3:42_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="hilbert3.html#E4:42_1_2_2"><i><font color="Green">E43</font></i></a>, <a class="txt" href="hilbert3.html#E7:42_1_2_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="hilbert3.html#E8:42_1_2_2"><i><font color="Green">E46</font></i></a></i>;<br/>

<b>thus </b><a NAME="E11:42_1_2_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="msualg_3.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>5</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbert3.html#E1:42_1_2_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="hilbert3.html#E3:42_1_2_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="hilbert3.html#E5:42_1_2_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="hilbert3.html#E7:42_1_2_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="hilbert3.html#E9:42_1_2_2"><i><font color="Green">E47</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:42_1_2"/>
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#E7:42_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="hilbert3.html#E8:42_1_2"><i><font color="Green">E39</font></i></a>, <a class="txt" href="hilbert3.html#E9:42_1_2"><i><font color="Green">E40</font></i></a>);<br/></i>
<a NAME="E11:42_1_2"/><b>then </b>
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="hilbert1.html#K1">HP-WFF</a>  holds <br/><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>
 
;<br/>
<b>hence </b><a NAME="E12:42_1_2"/>
<font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="pboole.html#T3">PBOOLE:3</a></i>;<br/>


</div>
