<?xml version="1.0"?>
<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="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  &amp; <font color="Maroon">a<sub>4</sub></font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  implies  ex <font color="Olive">p'</font>, <font color="Olive">q'</font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  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">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="trees_4.html#K10">-tree</a> <font color="Olive">p'</font>,<font color="Olive">q'</font> ) ) &amp; ( (  not <font color="Maroon">a<sub>3</sub></font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  or  not <font color="Maroon">a<sub>4</sub></font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  ) 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="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  &amp; <font color="Maroon">a<sub>4</sub></font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  implies  ex <font color="Olive">p'</font>, <font color="Olive">q'</font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  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">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="hilbert1.html#K3">=&gt;</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="trees_4.html#K10">-tree</a> <font color="Olive">p'</font>,<font color="Olive">q'</font> ) ) &amp; ( (  not <font color="Maroon">a<sub>3</sub></font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  or  not <font color="Maroon">a<sub>4</sub></font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  ) 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="E1:41_1_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>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_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="41_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="E1:41_1_1_1_1"/>
( ( <font color="Maroon">a</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  &amp; <font color="Maroon">b</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  ) or  not <font color="Maroon">a</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  or  not <font color="Maroon">b</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:41_1_1_1_1_1"/><i><font color="Green">E30</font></i>: 
<font color="Maroon">a</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a> 
 <b>and </b><br/><a NAME="E2:41_1_1_1_1_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">b</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">a</font> as   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">b</font> as   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  <b>by </b><i><a class="txt" href="hilbert2.html#E2:41_1_1_1_1_1"><i><font color="Green">E33</font></i></a></i>;<br/><b>take </b>
<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="trees_4.html#K10">-tree</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
;<br/><b>thus </b><a NAME="E5:41_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">(<span class="default"><font color="Maroon">p</font> <a href="hilbert1.html#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="trees_4.html#K10">-tree</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_1_1_2"/>
(  not <font color="Maroon">a</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  or  not <font color="Maroon">b</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_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="E2:41_1_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> 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="41_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="41_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="E1:41_1_1_2_1"/>
( ( <font color="Maroon">a</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  &amp; <font color="Maroon">b</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  ) or  not <font color="Maroon">a</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  or  not <font color="Maroon">b</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_1_1"><b>suppose </b></a><b>that </b><a NAME="E1:41_1_1_2_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">a</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a> 
 <b>and </b><br/><a NAME="E2:41_1_1_2_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">b</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a> 
 ;<br/><div class="add"><b>reconsider </b><font color="Maroon">p'</font> = <font color="Maroon">a</font> as   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">q'</font> = <font color="Maroon">b</font> as   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  <b>by </b><i><a class="txt" href="hilbert2.html#E2:41_1_1"><i><font color="Green">E34</font></i></a></i>;<br/><b>take </b>
<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="trees_4.html#K10">-tree</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>
;<br/><b>thus </b><a NAME="E5:41_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>,<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="trees_4.html#K10">-tree</a> <font color="Maroon">p'</font>,<font color="Maroon">q'</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_2_1_2"/>
(  not <font color="Maroon">a</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  or  not <font color="Maroon">b</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_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="E3:41_1_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> <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="E4:41_1_1"/><i><font color="Green">E39</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#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="trees_3.html#M5">Element</a> of  <a href="trees_3.html#K5">FinTrees</a> <a href="hilbert1.html#K1">HP-WFF</a>  =  <a href="trees_4.html#K2">root-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<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="E6:41_1_1"/><i><font color="Green">E40</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="trees_4.html#K2">root-tree</a> <a href="hilbert1.html#K2">VERUM</a> 
 <b>and </b><br/><a NAME="E7:41_1_1"/><i><font color="Green">E41</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">H<sub>1</sub></font>(<font color="Olive">n</font>)
 <b>and </b><br/><a NAME="E8:41_1_1"/><i><font color="Green">E77</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#E2:41_1_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="E9:41_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="trees_4.html#K2">root-tree</a> <a href="hilbert1.html#K2">VERUM</a> 
 <b>by </b><i/>;<br/>

<b>thus </b><a NAME="E10:41_1_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>  <a href="trees_4.html#K2">root-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font></span>)</span>
 <b>by </b><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>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="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a> ;<br/>
<a NAME="E11:41_1_1"/><i><font color="Green">E85</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[ <a href="hilbert1.html#K2">VERUM</a> ]
 
<b>by </b><i/>;<br/>
<a NAME="E12:41_1_1"/><i><font color="Green">E86</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">S<sub>3</sub></font>[ <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Olive">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:41_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" target="_self">prop</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="trees_4.html#K2">root-tree</a> <span class="p1">(<span class="default"><a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font></span>)</span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:41_1_1_3"/>
<font color="Maroon">S<sub>3</sub></font>[ <a href="hilbert2.html#K1" target="_self">prop</a> <font color="Maroon">n</font>]
 ;<br/>


</div><b>end;</b></div>
<a NAME="E13:41_1_1"/><i><font color="Green">E87</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="41_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="E1:41_1_1_4"/><i><font color="Green">E88</font></i>: 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a> 
 <b>and </b><br/><a NAME="E2:41_1_1_4"/><i><font color="Green">E89</font></i>: 
<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font> is   <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a> 
 ;<br/>

<a NAME="E3:41_1_1_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,<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>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E4:41_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="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a> 
 <b>by </b><i><a class="txt" href="hilbert2.html#E7:41_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="hilbert2.html#T2" target="_self">Th2</a></i>;<br/>

<a NAME="E5:41_1_1_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">r</font>,<font color="Maroon">s</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">s</font>,<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>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E6:41_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="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a> 
 <b>by </b><i><a class="txt" href="hilbert2.html#E7:41_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="hilbert2.html#T2" target="_self">Th2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E14:41_1_1"/><i><font color="Green">E90</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" target="_self">HILBERT2:sch 2</a>(<a class="txt" href="hilbert2.html#E3:41_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="hilbert2.html#E4:41_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="hilbert2.html#E6:41_1_1"><i><font color="Green">E40</font></i></a>);<br/></i>
<a NAME="E15:41_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>,<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>] &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">p</font>,<font color="Maroon">q</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font>,<font color="Maroon">M</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">q</font>,<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>] )
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E16:41_1_1"/>
 ex <font color="Olive">p'</font>, <font color="Olive">q'</font> being  <a href="trees_2.html#NM6">DecoratedTree</a> of  <a href="hilbert1.html#K1">HP-WFF</a>  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#K4">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</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="trees_4.html#K10">-tree</a> <font color="Olive">p'</font>,<font color="Olive">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> <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="trees_4.html#K10">-tree</a> <font color="Olive">p'</font>,<font color="Olive">q'</font> )
 <b>by </b><i><a class="txt" href="hilbert2.html#E7:41_1_1"><i><font color="Green">E41</font></i></a></i>;<br/>


</div>
