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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> ] <b>means </b> <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c1">V</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>;<br/>
<a NAME="E1:47_1_1"/>
 <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c2">P</font>,<a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a>  <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c1">V</font>,<a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T33" target="_self" title="HILBERT3:th.33">Th33</a></i>;<br/>
<a NAME="E2:47_1_1"/><b>then </b><i><font color="Green" title="E32">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert1.html#K2" title="HILBERT1:func.2">VERUM</a> ]
 
;<br/>
<a NAME="E3:47_1_1"/><i><font color="Green" title="E33">A2</font></i>: 
for <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[ <a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Olive" title="b1">n</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" title="c4">n</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>

<a NAME="E1:47_1_1_1"/><i><font color="Green" title="E34">A3</font></i>: 
 <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">V</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/>
<a NAME="E2:47_1_1_1"/>
 <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c2">P</font>,<span class="p1">(<span class="default"><a href="hilbert2.html#K1" title="HILBERT2:func.1">prop</a> <font color="Maroon" title="c4">n</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">P</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">n</font>
 
<b>by </b><i><a class="ref" href="hilbert3.html#D5" target="_self" title="HILBERT3:def.5">Def5</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" title="HILBERT2:func.1">prop</a> <font color="Maroon" title="c4">n</font>]
 <b>by </b><i><a class="txt" href="hilbert3.html#E1:47_1_1_1"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="hilbert3.html#D4" target="_self" title="HILBERT3:def.4">Def4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:47_1_1"/><i><font color="Green" title="E34">A4</font></i>: 
for <font color="Olive" title="b1">r</font>, <font color="Olive" title="b2">s</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">s</font>] holds <br/>( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Olive" title="b2">s</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Olive" title="b2">s</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" title="c4">r</font>, <font color="Maroon" title="c5">s</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a> ;<br/>



<a NAME="E1:47_1_1_2"/><i><font color="Green" title="E35">A5</font></i>: 
 <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c4">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c5">s</font></span>)</span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/>
<a NAME="E2:47_1_1_2"/><i><font color="Green" title="E36">A6</font></i>: 
 <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c1">V</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fraenkel.html#K1" title="FRAENKEL:func.1">Funcs</a> <span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c4">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c5">s</font></span>)</span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#D2" target="_self" title="HILBERT3:def.2">Def2</a></i>;<br/>
<b>assume </b><a NAME="E3:47_1_1_2"/>
 <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c4">r</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">r'</font> =  <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c4">r</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c4">r</font> ;<br/>
<b>assume </b><a NAME="E5:47_1_1_2"/>
 <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">s'</font> =  <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c5">s</font> as   <a href="funct_2.html#NM2" title="FUNCT_2:NM.2">Permutation</a> of  <a href="hilbert3.html#K3" title="HILBERT3:func.3">SetVal</a> <font color="Maroon" title="c1">V</font>,<font color="Maroon" title="c5">s</font> ;<br/>
<a NAME="E7:47_1_1_2"/>
 <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c2">P</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <span class="p1"><a href="funct_3.html#K16" title="FUNCT_3:func.16">[:</a><span class="default"><font color="Maroon" title="c6">r'</font>,<font color="Maroon" title="c7">s'</font></span><a href="funct_3.html#K16" title="FUNCT_3:func.16">:]</a></span>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T35" target="_self" title="HILBERT3:th.35">Th35</a></i>;<br/>
<b>hence </b><a NAME="E8:47_1_1_2"/>
 <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c2">P</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="hilbert1.html#K4" title="HILBERT1:func.4">'&amp;'</a> <font color="Maroon" title="c5">s</font></span>)</span> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 <b>by </b><i><a class="txt" href="hilbert3.html#E1:47_1_1_2"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="hilbert3.html#T25" target="_self" title="HILBERT3:th.25">Th25</a></i>;<br/>

<a NAME="E9:47_1_1_2"/>
 <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c2">P</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c5">s</font></span>)</span> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c6">r'</font> <a href="hilbert3.html#K1" title="HILBERT3:func.1">=&gt;</a> <font color="Maroon" title="c7">s'</font>
 
<b>by </b><i><a class="ref" href="hilbert3.html#T36" target="_self" title="HILBERT3:th.36">Th36</a></i>;<br/>
<b>hence </b><a NAME="E10:47_1_1_2"/>
 <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c2">P</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="hilbert1.html#K3" title="HILBERT1:func.3">=&gt;</a> <font color="Maroon" title="c5">s</font></span>)</span> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 <b>by </b><i><a class="txt" href="hilbert3.html#E2:47_1_1_2"><i><font color="Green" title="E36">A6</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:47_1_1"/>
for <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="hilbert1.html#K1" title="HILBERT1:func.1">HP-WFF</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 
<b>from </b><i><a class="ref" href="hilbert2.html#S2" title="HILBERT2:sch.2">HILBERT2:sch 2</a>(<a class="txt" href="hilbert3.html#E2:47_1_1"><i><font color="Green" title="E32">A1</font></i></a>, <a class="txt" href="hilbert3.html#E3:47_1_1"><i><font color="Green" title="E33">A2</font></i></a>, <a class="txt" href="hilbert3.html#E4:47_1_1"><i><font color="Green" title="E34">A4</font></i></a>);<br/></i>
<b>hence </b><a NAME="E6:47_1_1"/>
 <a href="hilbert3.html#K5" title="HILBERT3:func.5">Perm</a> <font color="Maroon" title="c2">P</font>,<font color="Maroon" title="c3">p</font> is <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a>
 ;<br/>


</div>
